Domain of Real Natural Logarithm

Let $\ln$ be the natural logarithm function on the real numbers. Then the domain of $\ln$ is the set of strictly positive real numbers: :$\Dom \ln = \R_{>0}$


Image of Real Natural Logarithm

Let $\ln$ be the natural logarithm function on the real numbers. Then the image of $\ln$ is the set of real numbers: :$\Img \ln = \R$


Soundness and Completeness of Gentzen Proof System

Let $\mathscr G$ be instance 1 of a Gentzen proof system. Let $\mathrm{BI}$ be the formal semantics of boolean interpretations. Then $\mathscr G$ is a sound and complete proof system for $\mathrm{BI}$.


Definition:Deduction Rule

Let $\mathcal L$ be the language of propositional logic. The '''Deduction Rule''' is the rule of inference: :From $U, \mathbf A \vdash \mathbf B$, one may infer $U \vdash \mathbf A \implies \mathbf B$ where: :$\mathbf A, \mathbf B$ are WFFs of propositional logic :$U$ is a set of WFFs For a given proof system, the '''Deduction Rule''' can be either a basic rule of inference, or a derived rule.


Deduction Theorem

Let $\mathscr H$ be instance 1 of a Hilbert proof system. Then the deduction rule: ::$\dfrac{U,\mathbf A \vdash \mathbf B}{U \vdash \mathbf A \implies \mathbf B}$ is a derived rule for $\mathscr H$.


Soundness Theorem for Hilbert Proof System

Let $\mathscr H$ be instance 1 of a Hilbert proof system. Let $\mathrm{BI}$ be the formal semantics of boolean interpretations. Then $\mathscr H$ is a sound proof system for $\mathrm{BI}$: :Every $\mathscr H$-theorem is a tautology.


Equivalence of Definitions of Dominate (Set Theory)

Let $S, T$ be sets. {{TFAE|def = Dominate (Set Theory)|view = Dominate|context = Set Theory}}


Element of Toset has at most One Immediate Successor

Let $\struct {S, \preceq}$ be a toset. Let $a \in S$. Then $a$ has at most one immediate successor.


Element of Toset has at most One Immediate Predecessor

Let $\struct {S, \preceq}$ be a toset. Let $a \in S$. Then $a$ has at most one immediate predecessor.


Equivalence of Definitions of Complex Inverse Tangent Function

{{TFAE|def = Complex Inverse Tangent}} Let $S$ be the subset of the complex plane: :$S = \C \setminus \set {0 + i, 0 - i}$


Equivalence of Definitions of Complex Inverse Cotangent Function

{{TFAE|def = Complex Inverse Cotangent}} Let $S$ be the subset of the complex plane: :$S = \C \setminus \left\{{0 + i, 0 - i}\right\}$


Square Root of Complex Number in Cartesian Form

Let $z \in \C$ be a complex number. Let $z = x + i y$ where $x, y \in \R$ are real numbers. Let $z$ not be wholly real, that is, such that $y \ne 0$. Then the square root of $z$ is given by: :$z^{1/2} = \pm \paren {a + i b}$ where: {{begin-eqn}} {{eqn | l = a       | r = \sqrt {\frac {x + \sqrt {x^2 + y^2} } 2}       | c =  }} {{eqn | l = b       | r = \frac y {\cmod y} \sqrt {\frac {-x + \sqrt {x^2 + y^2} } 2}       | c =  }} {{end-eqn}}


Zero is Identity in Naturally Ordered Semigroup

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $0$ be the zero of $\struct {S, \circ, \preceq}$. Then $0$ is the identity for $\circ$. That is: :$\forall n \in S: n \circ 0 = n = 0 \circ n$


Reduction Formula for Integral of Power of Sine

Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Then: :$\displaystyle \int \sin^n x \rd x = \dfrac {n - 1} n \int \sin^{n - 2} x \rd x - \dfrac {\sin^{n - 1} x \cos x} n$ is a reduction formula for $\displaystyle \int \sin^n x \rd x$.


Reduction Formula for Definite Integral of Power of Sine

Let $n \in \Z_{> 0}$ be a positive integer. Let $I_n$ be defined as: :$\displaystyle I_n = \int_0^{\frac \pi 2} \sin^n x \rd x$ Then $\left\langle{I_n}\right\rangle$ is a decreasing sequence of real numbers which satisfies: :$n I_n = \left({n - 1}\right) I_{n - 2}$ Thus: :$I_n = \dfrac {n - 1} n  I_{n - 2}$ is a reduction formula for $I_n$.


Stirling's Formula/Proof 2/Lemma 4

Let $I_n$ be defined as: :$\displaystyle I_n = \int_0^{\frac \pi 2} \sin^n x \rd x$ Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac {I_{2 n} } {I_{2 n + 1} } = 1$


Definite Integral from 0 to Half Pi of Even Power of Sine x

:$\displaystyle \int_0^{\frac \pi 2} \sin^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$


Definite Integral from 0 to Half Pi of Odd Power of Sine x

:$\displaystyle \int_0^{\frac \pi 2} \sin^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$


Zero Complement is Not Empty

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $S^*$ be the zero complement of $S$. Then $S^*$ is not empty.


Stirling's Formula/Proof 2/Lemma 5

:$\dfrac {n!} {n^n \sqrt n e^{-n} } \to \sqrt {2 \pi}$ as $n \to \infty$


Limit of Error in Stirling's Formula

Consider Stirling's Formula: :$n! \sim \sqrt {2 \pi n} \left({\dfrac n e}\right)^n$ The ratio of $n!$ to its approximation $\sqrt {2 \pi n} \left({\dfrac n e}\right)^n$ is bounded as follows: :$e^{1 / \left({12 n + 1}\right)} \le \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } \le e^{1 / 12 n}$


Zero Strictly Precedes One

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $0$ be the zero of $S$. Let $1$ be the one of $S$. Then: :$0 \prec 1$


Stirling's Formula/Proof 2/Lemma 2

The sequence $\left \langle {d_n} \right \rangle$ defined as: : $\displaystyle d_n = \ln \left({n!}\right) - \left({n + \frac 1 2}\right) \ln n + n$ is decreasing.


Power Series Expansion for Real Arctangent Function

The arctangent function has a Taylor series expansion: :$\arctan x = \begin {cases} \displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \displaystyle \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \ge 1 \\ \displaystyle -\frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \le -1 \end {cases}$ That is: :$\arctan x = \begin {cases} \displaystyle x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots & : -1 \le x \le 1 \\ \displaystyle \frac \pi 2 - \frac 1 x + \frac 1 {3 x^3} - \frac 1 {5 x^5} + \cdots & : x \ge 1 \\ \displaystyle -\frac \pi 2 - \frac 1 x + \frac 1 {3 x^3} - \frac 1 {5 x^5} + \cdots & : x \le -1 \end {cases}$


Ordering in terms of Addition

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Then $\forall m, n \in S$:  :$m \preceq n \iff \exists p \in S: m \circ p = n$


Difference in Naturally Ordered Semigroup is Unique

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $n, m \in S$ such that $m \preceq n$. Then there exists a unique difference $n - m$ of $m$ and $n$.


Equivalence of Definitions of Beta Function

{{TFAE|def = Beta Function}} For $\map \Re x, \map \Re y > 0$:


Ordering of Naturally Ordered Semigroup is Strongly Compatible

:$\forall m, n, p \in S: m \preceq n \iff m \circ p \preceq n \circ p$


Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible

:$\forall m, n, p \in S: m \prec n \iff m \circ p \prec n \circ p$


Sum with One is Immediate Successor in Naturally Ordered Semigroup

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $1$ be the one of $S$. Let $n \in S$. Then $n \circ 1$ is the immediate successor of $n$. That is, for all $m \in S$: :$n \prec m \iff n \circ 1 \preceq m$


Riemann Zeta Function at Even Integers

The Riemann $\zeta$ function can be calculated for even integers as follows: {{begin-eqn}} {{eqn | l = \map \zeta {2 n}       | r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = \frac 1 {1^{2 n} } + \frac 1 {2^{2 n} } + \frac 1 {3^{2 n} } + \frac 1 {4^{2 n} } + \cdots       | c =  }} {{end-eqn}} where: :$B_n$ are the Bernoulli numbers :$n$ is a positive integer.


Tangent Inequality

:$x < \tan x$ for all $x$ in the interval $\left({0 \,.\,.\, \dfrac {\pi} 2}\right)$.


Integral Representation of Riemann Zeta Function in terms of Gamma Function

For $\Re \paren s > 1$, the Riemann Zeta function is given by: :$\displaystyle \map \zeta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {t^{s - 1}} {e^t - 1} \rd t$ where $\Gamma$ is the Gamma function.


Gamma Function is Unique Extension of Factorial

Let $f: \R_{>0} \to \R$ be a real function which is positive and continuous. Let $\ln \mathop \circ f$ be convex on $\R_{>0}$. Let $f$ satisfy the conditions: :$f \left({x + 1}\right) = \begin{cases} 1 & : x = 0 \\ x f \left({x}\right) & : x > 0 \end{cases}$ Then: : $\forall x \in \R_{>0}: f \left({x}\right) = \Gamma \left({x}\right)$ where $\Gamma \left({x}\right)$ is the Gamma function.


Equivalence of Definitions of Convex Real Function

Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Convex Real Function}}


Power Series Expansion for Real Arcsine Function

The (real) arcsine function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \arcsin x       | r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}       | c =  }} {{eqn | r = x + \frac 1 2 \frac {x^3} 3 + \frac {1 \times 3} {2 \times 4} \frac {x^5} 5 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} \frac {x^7} 7 + \cdots       | c =  }} {{end-eqn}} which converges for $-1 \le x \le 1$.


Power Series Expansion for Real Arccosine Function

The arccosine function has a Taylor Series expansion: {{begin-eqn}} {{eqn | l = \arccos x       | r = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}       | c =  }} {{eqn | r = \frac \pi 2 - \paren {x + \frac 1 2 \frac {x^3} 3 + \frac {1 \times 3} {2 \times 4} \frac {x^5} 5 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} \frac {x^7} 7 + \cdots}       | c =  }} {{end-eqn}} which converges for $-1 \le x \le 1$.


Real Function is Concave iff its Negative is Convex

Let $f$ be a real function. Let $I \subseteq \R$ be an interval of $\R$. Then $f$ is concave on $I$ {{iff}} $-f$ is convex on $\R$.


Concave Real Function is Left-Hand and Right-Hand Differentiable

Let $f$ be a real function which is concave on the open interval $\left({a \,.\,.\, b}\right)$. Then the left-hand derivative $f'_- \left({x}\right)$ and right-hand derivative $f'_+ \left({x}\right)$ both exist for all $x \in \left({a \,.\,.\, b}\right)$.


Absolute Value Function is Convex

Let $f: \R \to \R$ be the absolute value function on the real numbers. Then $f$ is convex.


Real Function is Concave iff Derivative is Decreasing

Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Then $f$ is concave on $\openint a b$ {{iff}} its derivative $f'$ is decreasing on $\openint a b$. Thus the intuitive result that a concave function "gets less steep".


Real Function is Strictly Convex iff Derivative is Strictly Increasing

Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Then $f$ is strictly convex on $\openint a b$ {{iff}} its derivative $f'$ is strictly increasing on $\openint a b$.


Real Function is Strictly Concave iff Derivative is Strictly Decreasing

Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Then $f$ is strictly concave on $\openint a b$ {{iff}} its derivative $f'$ is strictly decreasing on $\openint a b$.


Inverse of Strictly Decreasing Convex Real Function is Convex

Let $f$ be a real function which is convex on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is convex on $J$.


Inverse of Strictly Increasing Concave Real Function is Convex

Let $f$ be a real function which is concave on the open interval $I$. Let $J = f \sqbrk I$. If $f$ is strictly increasing on $I$, then $f^{-1}$ is convex on $J$.


Inverse of Strictly Decreasing Concave Real Function is Concave

Let $f$ be a real function which is concave on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is concave on $J$.


Mean Value of Concave Real Function

Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Let $f$ be concave on $\left({a \,.\,.\, b}\right)$. Then: : $\forall \xi \in \left({a \,.\,.\, b}\right): f \left({x}\right) - f \left({\xi}\right) \le f^{\prime} \left({\xi}\right) \left({x - \xi}\right)$


Differentiable Bounded Concave Real Function is Constant

Let $f$ be a real function which is: :$(1): \quad$ Differentiable on $\R$ :$(2): \quad$ Bounded on $\R$ :$(3): \quad$ Concave on $\R$. Then $f$ is constant.


Second Derivative of Strictly Concave Real Function is Strictly Negative

Let $f$ be a real function which is twice differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Then $f$ is strictly concave on $\left({a \,.\,.\, b}\right)$ {{iff}} its second derivative $f'' < 0$ on $\left({a \,.\,.\, b}\right)$.


Second Derivative of Strictly Convex Real Function is Strictly Positive

Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$. Then $f$ is strictly convex on $\openint a b$ {{iff}} its second derivative $f'' > 0$ on $\openint a b$.


Inverse of Strictly Decreasing Strictly Concave Real Function is Strictly Concave

Let $f$ be a real function which is strictly concave on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is strictly concave on $J$.


Inverse of Strictly Increasing Strictly Convex Real Function is Strictly Concave

Let $f$ be a real function which is strictly convex on the open interval $I$. Let $J = f \sqbrk I$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is strictly concave on $J$.


Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex

Let $f$ be a real function which is strictly concave on the open interval $I$. Let $J = f \left[{I}\right]$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is strictly convex on $J$.


Gamma Function as Integral of Natural Logarithm

Let $x \in \R_{>0}$ be a strictly positive real number. Then: :$\displaystyle \map \Gamma x = \int_{\to 0}^1 \paren {\ln \frac 1 t}^{x - 1} \rd t$ where $\Gamma$ denotes the Gamma function.


Reversal of Limits of Definite Integral

Let $a \le b$. Then: :$\displaystyle \int_a^b \map f x \rd x = - \int_b^a \map f x \rd x$


Comparison Test for Improper Integral

Let $I = \openint a b$ be an open real interval. Let $\phi$ be a real function which is continuous on $I$ and also non-negative on $I$. Let $f$ be a real function which is continuous on $I$. Let $f$ satisfy: :$\forall x \in I: \size {\map f x} \le \map \phi x$ If the improper integral of $\phi$ over $I$ exists, then so does that of $f$.


Beta Function is Continuous and Positive on Positive Reals

Let $x, y \in \R$ be real numbers. Let $\map \Beta {x, y}$ be the Beta function: :$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$ Let $y \in \R_{>0}$ be given. Then $\map \Beta {x, y}$ is a positive and continuous function of $x$ on $\R_{>0}$.


Logarithm of Beta Function is Convex on Positive Reals

Let $x, y \in \R$ be real numbers. Let $\Beta \left({x, y}\right)$ be the Beta function: :$\displaystyle \Beta \left({x, y}\right) := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \left({1 - t}\right)^{y - 1} \ \mathrm d t$ Let $y \in \R_{>0}$ be given. Then $\ln \left({\Beta \left({x, y}\right)}\right)$ is a convex function of $x$ on $\R_{>0}$.


Gamma Function of One Half

:$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ Its decimal expansion starts: :$\map \Gamma {\dfrac 1 2} = 1 \cdotp 77245 \, 38509 \, 05516 \, 02729 \, 81674 \, 83341 \, 14518 \, 27975 \ldots$


Sum of Reciprocals of Primes is Divergent/Proof 2

The series: :$\displaystyle \sum_{p \mathop \in \Bbb P} \frac 1 p$ where: :$\Bbb P$ is the set of all prime numbers is divergent.


Sum of Reciprocals of Primes is Divergent/Lemma

Let $C \in \R_{>0}$ be a (strictly) positive real number. Then: :$\displaystyle \lim_{n \mathop \to \infty} \paren {\map \ln {\ln n} - C} = + \infty$


Strictly Increasing Infinite Sequence of Integers is Cofinal in Natural Numbers

Let $S = \left\langle{x_n}\right\rangle$ be an infinite sequence of integers which is strictly increasing.   Then $S$ is a cofinal subset of $\left({\Z, \le}\right)$ where $\le$ is the usual ordering on the integers.


Product of Even and Odd Functions

Let $\mathcal O$ be an odd real function defined on some symmetric set $S$. Let $\mathcal E$ be an even real function defined on some symmetric set $S'$. Let $\mathcal O\mathcal E$ be their pointwise product, defined on the intersection of the domains of $\mathcal O$ and $\mathcal E$. Then $\mathcal O\mathcal E$ is odd.  That is: :$\forall x \in S \cap S': \left({\mathcal O\mathcal E}\right)\left({-x}\right) = - \left({\mathcal O\mathcal E}\right)\left({x}\right)$.


Millin Series

The '''Millin series''' is the series defined as: :$\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {F_{2^n}} = \frac {7 - \sqrt 5} 2$


Odd Bernoulli Numbers Vanish

Let $B_n$ denote the $n$th Bernoulli Number. Then: :$B_{2n + 1} = 0$ for $n \ge 1$.


Gauss Multiplication Formula

Let $\Gamma$ denote the Gamma Function. Then: :$\displaystyle \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$ where $\N$ denotes the natural numbers.


Dirichlet Integral

:$\displaystyle \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$


Riemann Zeta Function and Prime Counting Function

For $\map \Re s > 1$: :$\displaystyle \log \map \zeta s = s \int_0^{\mathop \to \infty} \frac {\map \pi x} {x \paren {x^s - 1} } \rd x$ where: :$\zeta$ denotes the Riemann Zeta Function :$\pi$ denotes the Prime-Counting Function.


Faulhaber's Formula

Let $n$ and $p$ be positive integers. Then: :$\displaystyle \sum_{k \mathop = 1}^n k^p = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}$ where $B_n$ denotes the $n$th Bernoulli number.


Complex Exponential Tends to Zero

Let $\exp z$ be the complex exponential function. Then: :$\displaystyle \lim_{\map \Re z \mathop \to +\infty} e^{-z} = 0$ where $\map \Re z$ denotes the real part of $z$.


Modulus of Exponential is Exponential of Real Part

Let $z \in \C$ be a complex number. Let $\exp z$ denote the complex exponential function. Let $\cmod {\, \cdot \,}$ denote the complex modulus Then: :$\cmod {\exp z} = \map \exp {\Re z}$ where $\Re z$ denotes the real part of $z$.


Laplace Transform of Exponential/Real Argument

Let $\laptrans f$ denote the Laplace transform of a function $f$. Let $e^x$ be the real exponential. Then: :$\map {\laptrans {e^{a t} } } s = \dfrac 1 {s - a}$ where $a \in \R$ is constant, and $\map \Re s > \map \Re a$.


Laplace Transform of Cosine

Let $\cos$ be the real cosine function. Let $\laptrans f$ denote the Laplace transform of the real function $f$. Then: :$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$ where $a \in \R_{>0}$ is constant, and $\map \Re s > a$.


Laplace Transform of Sine

Let $\sin$ denote the real sine function. Let $\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$ where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$.


Sum Rule for Derivatives/General Result

Let $f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)$ be real functions all differentiable. Then for all $n \in \N_{>0}$: : $\displaystyle D_x \left({\sum_{i \mathop = 1}^n f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^n D_x \left({f_i \left({x}\right)}\right)$


Product Rule for Derivatives/General Result

Let $f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)$ be real functions differentiable on the open interval $I$. then: :$\forall x \in I: \displaystyle D_x \left({\prod_{i \mathop = 1}^n f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^n \left({D_x \left({f_i \left({x}\right)}\right) \prod_{j \mathop \ne i} f_j \left({x}\right)}\right)$


Derivative of Arccotangent Function

:$\dfrac {\map \d {\arccot x} } {\d x} = \dfrac {-1} {1 + x^2}$


Asymptotic Formula for Bernoulli Numbers

The Bernoulli numbers with even index can be approximated by the asymptotic formula: :$B_{2 n} \sim \paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\dfrac n {\pi e} }^{2 n}$ where: :$B_n$ denotes the $n$th Bernoulli number :$\sim$ denotes asymptotically equal.


Cotangent Minus Tangent

:$\cot x - \tan x = 2 \cot 2 x$


Derivative of Hyperbolic Cotangent Function

:$\map {\dfrac \d {\d x} } {\coth u} = -\csch^2 u \dfrac {\d u} {\d x}$


Derivative of Hyperbolic Cosecant Function

:$\map {\dfrac \d {\d x} } {\csch u} = -\csch u \coth u \dfrac {\d u} {\d x}$


Derivative of Inverse Hyperbolic Sine

:$\map {\dfrac \d {\d x} } {\sinh^{-1} x} = \dfrac 1 {\sqrt {x^2 + 1} }$


Derivative of Inverse Hyperbolic Cosine

:$\map {\dfrac \d {\d x} } {\cosh^{-1} x} = \dfrac 1 {\sqrt {x^2 - 1} }$


Derivative of Inverse Hyperbolic Tangent

:$\map {\dfrac \d {\d x} } {\tanh^{-1} x} = \dfrac 1 {1 - x^2}$


Derivative of Inverse Hyperbolic Cotangent

:$\map {\dfrac \d {\d x} } {\coth^{-1} x} = \dfrac {-1} {x^2 - 1}$


Derivative of Inverse Hyperbolic Secant

:$\map {\dfrac \d {\d x} } {\sech^{-1} x} = \dfrac {-1} {x \sqrt{1 - x^2} }$


Derivative of Inverse Hyperbolic Cosecant

:$\map {\dfrac \d {\d x} } {\csch^{-1} x} = \dfrac {-1} {\size x \sqrt {1 + x^2} }$


Sum of Cosecant and Cotangent

:$\displaystyle \csc x + \cot x = \cot {\frac x 2}$


Leibniz's Rule/One Variable/Third Derivative

Let $f$ and $g$ be real functions defined on the open interval $I$. Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are thrice differentiable. Then: :$\paren {\map f x \, \map g x}''' = \map f x \, \map {g'''} x + 3 \map {f'} x \, \map {g''} x + 3 \map {f''} x \, \map {g'} x + \map {f'''} x \, \map g x$


Power Series Expansion for Tangent Function

The tangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \tan x       | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}       | c =  }} {{eqn | r = x + \frac {x^3} 3 + \frac {2 x^5} {15} + \frac {17 x^7} {315} + \cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $\size x < \dfrac \pi 2$.


Partial Differentiation Operator is Commutative for Continuous Functions

Let $\map f {x, y}$ be a function of the two independent variables $x$ and $y$. Let $\map f {x, y}$ be continuous. Let the partial deriviatives of $f$ also be continuous. Then: :$\dfrac {\partial^2 f} {\partial x \partial y} = \dfrac {\partial^2 f} {\partial y \partial x}$


Primitive of Constant Multiple of Function

Let $f$ be a real function which is integrable. Let $c$ be a constant. Then: :$\displaystyle \int c \map f x \rd x = c \int \map f x \rd x$


Primitive of Pointwise Sum of Functions

Let $f_1, f_2, \ldots, f_n$ be real functions which are integrable. Then: :$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$


Hurwitz's Theorem (Number Theory)

Let $\xi$ be an irrational number. Then there are infinitely many relatively prime integers $m, n \in \Z$ such that: :$\left|{\xi - \dfrac p q}\right| < \dfrac 1 {\sqrt 5 \, q^2}$


Primitive of Function of Constant Multiple

Let $f$ be a real function which is integrable. Let $c$ be a constant. Then: :$\displaystyle \int \map f {c x} \rd x = \frac 1 c \int \map f u \d u$ where $u = c x$.


Primitive of Composite Function

Let $f$ and $g$ be a real functions which are integrable. Let the composite function $g \circ f$ also be integrable. Then: {{begin-eqn}} {{eqn | l = \int \map {\paren {g \circ f} } x \rd x       | r = \int \map g u \frac {\d x} {\d u} \rd u       | c =  }} {{eqn | r = \int \frac {\map g u} {\map {f'} x} \rd u       | c =  }} {{end-eqn}} where $u = \map f x$.


Weierstrass Product Inequality

For $n \ge 1$: :$\displaystyle \prod_{i \mathop = 1}^n \left({1 - a_i}\right) \ge 1 - \sum_{i \mathop = 1}^n a_i$ where all of $a_i$ are in the closed interval $\left[{0 \,.\,.\,1 }\right]$.


Primitive of Power

Let $n \in \R: n \ne -1$. Then: :$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$ where $C$ is an arbitrary constant. That is: :$\dfrac {x^{n + 1} } {n + 1}$ is a primitive of $x^n$.


Sine Inequality

:$\size {\sin x} \le \size x$ for all $x \in \R$.


Jordan's Inequality

:$\dfrac 2 {\pi} x \le \sin x \le x$ for all $x$ in the interval $\left[{0 \,.\,.\, \dfrac {\pi} 2}\right]$


Cosine Inequality

:$1 - \dfrac {x^2} 2 \le \cos x$ for all $x \in \R$.


Laplace Transform of Constant Mapping

Let $a \in \R$ be a real constant. Let $f_a: \R \to \R$ or $\C$ be the constant mapping, defined as: :$\forall t \in \R: \map {f_a} t = a$ Let $\laptrans {f_a}$ be the Laplace transform of $f_a$. Then: :$\laptrans {\map {f_a} t} = \dfrac a s$ for $\map \Re s > a$.


Laplace Transform of Positive Integer Power

Let $\laptrans f$ denote the Laplace transform of a function $f$. Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$. Then: :$\laptrans {t^n} = \dfrac {n!} { s^{n + 1} }$ for $\map \Re s > 0$.


Primitive of Function under its Derivative

Let $f$ be a real function which is integrable. Then: :$\ds \int \frac {\map {f'} x} {\map f x} \rd x = \ln \size {\map f x} + C$ where $C$ is an arbitrary constant.


Primitive of Exponential Function/General Result

Let $a \in \R_{>0}$ be a constant such that $a \ne 1$. Then: :$\ds \int a^x \rd x = \frac {a^x} {\ln a} + C$ where $C$ is an arbitrary constant.


Primitive of Tangent Function/Secant Form

:$\ds \int \tan x \rd x = \ln \size {\sec x} + C$ where $\sec x$ is defined.


Primitive of Secant Function/Tangent plus Angle Form

:$\ds \int \sec x \rd x = \ln \size {\map \tan {\frac x 2 + \frac \pi 4} } + C$


Exponential of x not less than 1+x

:$e^x \ge 1 + x$ for all $x \in \R$.


Primitive of Cosecant Function/Cosecant minus Cotangent Form

:$\ds \int \csc x \rd x = \ln \size {\csc x - \cot x} + C$ where $\csc x - \cot x \ne 0$.


Primitive of Cosecant Function/Tangent Form

:$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$ where $\tan \dfrac x 2 \ne 0$.


Primitive of Square of Secant Function

:$\ds \int \sec^2 x \rd x = \tan x + C$ where $C$ is an arbitrary constant.


Primitive of Square of Cosecant Function

:$\ds \int \csc^2 x \rd x = -\cot x + C$ where $C$ is an arbitrary constant.


Primitive of Square of Tangent Function

:$\displaystyle \int \tan^2 x \rd x = \tan x - x + C$


Primitive of Square of Cotangent Function

:$\displaystyle \int \cot^2 x \rd x = -\cot x - x + C$ where $C$ is an arbitrary constant.


Primitive of Square of Sine Function

:$\ds \int \sin^2 x \rd x = \frac x 2 - \frac {\sin 2 x} 4 + C$ where $C$ is an arbitrary constant.


Primitive of Square of Cosine Function

:$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$


Primitive of Product of Secant and Tangent

:$\ds \int \sec x \tan x \rd x = \sec x + C$ where $C$ is an arbitrary constant.


Primitive of Hyperbolic Sine Function

:$\ds \int \sinh x \rd x = \cosh x + C$


Primitive of Hyperbolic Cosine Function

:$\displaystyle \int \cosh x \rd x = \sinh x + C$


Primitive of Hyperbolic Tangent Function

:$\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$


Primitive of Hyperbolic Cotangent Function

:$\ds \int \coth x \rd x = \ln \size {\sinh x} + C$ where $\sinh x \ne 0$.


Limit of Tan X over X

:$\displaystyle \lim_{x \mathop \to 0} \frac {\tan x} x = 1$


Logarithm of Reciprocal

:$\map {\log_b} {\dfrac 1 x} = -\log_b x$


Primitive of Hyperbolic Secant Function/Arcsine Form

:$\ds \int \sech x \rd x = \map \arcsin {\tanh x} + C$


Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form

:$\ds \int \sech x \rd x = 2 \map \arctan {e^x} + C$


Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form

:$\ds \int \csch x \rd x = \ln \size {\tanh \frac x 2} + C$ where $\tanh \dfrac x 2 \ne 0$.


Primitive of Square of Hyperbolic Secant Function

:$\ds \int \sech^2 x \rd x = \tanh x + C$ where $C$ is an arbitrary constant.


Primitive of Square of Hyperbolic Cosecant Function

:$\ds \int \csch^2 x \rd x = -\coth x + C$ where $C$ is an arbitrary constant.


Primitive of Square of Hyperbolic Tangent Function

:$\displaystyle \int \tanh^2 x \rd x = x - \tanh x + C$ where $C$ is an arbitrary constant.


Primitive of Square of Hyperbolic Cotangent Function

:$\displaystyle \int \coth^2 x \rd x = x - \coth x + C$ where $C$ is an arbitrary constant.


Heaviside Step Function is Piecewise Continuous

Let $c \ge 0$ be a constant real number. The Heaviside step function: :$\mu_c \left({t}\right) = \begin{cases} 1 & : t > c \\ 0 & : t < c \end{cases}$ is piecewise continuous for any interval of the form: :$\left[{c - M \,.\,.\, c + M}\right]$ where $M > 0$ is some arbitrarily large constant.


Euler Formula for Sine Function/Complex Numbers/Proof 1/Lemma 1

The function: :$\dfrac {\sinh x} x$  is increasing for positive real $x$.


Hyperbolic Tangent Less than X

:$\tanh x \le x$ for $x \ge 0$.


Primitive of Square of Hyperbolic Sine Function

:$\displaystyle \int \sinh^2 x \rd x = \frac {\sinh 2 x} 4 - \frac x 2 + C$ where $C$ is an arbitrary constant.


Primitive of Square of Hyperbolic Cosine Function

:$\displaystyle \int \cosh^2 x \rd x = \frac {\sinh 2 x} 4 + \frac x 2 + C$ where $C$ is an arbitrary constant.


Primitive of Product of Hyperbolic Cosecant and Cotangent

:$\ds \int \csch x \coth x \rd x = -\csch x + C$ where $C$ is an arbitrary constant.


Primitive of Reciprocal of x squared minus a squared

thumbright600px$\color {blue} {\dfrac 1 {x^2 - a^2} } \qquad \color {green} {-\dfrac 1 a \tanh^{-1} \dfrac x a} \qquad \color {red} {-\dfrac 1 a \coth^{-1} \dfrac x a}$ Let $a \in \R_{>0}$ be a strictly positive real constant.


Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1

:$\displaystyle \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a\\ & \\  \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\ & \\ \text {undefined} & : \size x = a \end {cases}$


Primitive of Reciprocal of a squared minus x squared

thumbright600px$\color {blue} {\dfrac 1 {a^2 - x^2} } \qquad \color {red} {\dfrac 1 a \tanh^{-1} \dfrac x a} \qquad \color {green} {\dfrac 1 a \coth^{-1} \dfrac x a}$ Let $a \in \R_{>0}$ be a strictly positive real constant.


Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form

:$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$


Limit at Infinity of Polynomial over Complex Exponential

Let $n \in \N$. Let $\map {P_n} x$ be a real polynomial, of degree $n$. Let $e^z$ be the complex exponential function, where $z = x + i y$. Let $a \in \R_{>0}$. Then: :$\displaystyle \lim_{x \mathop \to +\infty} \frac {\map {P_n} x} {e^{a z} } = 0$


Primitives of Trigonometric Functions

This page gathers together primitives of trigonometric functions. In the below, $C$ is an arbitrary constant throughout.


Primitives of Hyperbolic Functions

This page gathers together primitives of hyperbolic functions. In the below, $C$ is an arbitrary constant throughout.


Derivatives of Trigonometric Functions

This page gathers together derivatives of trigonometric functions.


Derivatives of Inverse Trigonometric Functions

This page gathers together derivatives of inverse trigonometric functions.


Derivatives of Hyperbolic Functions

This page gathers together derivatives of hyperbolic functions. Let $u$ be a differentiable real function of $x$.


Derivatives of Inverse Hyperbolic Functions

This page gathers together derivatives of inverse hyperbolic functions.


Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form

:$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {x + \sqrt {x^2 - a^2} } + C$


Primitive of Reciprocal of x by Root of x squared minus a squared

:$\displaystyle \int \frac {\d x} {x \sqrt {x^2 - a^2} } = \frac 1 a \arcsec \size {\frac x a} + C$ for $\size x > a$.


Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form

:$\displaystyle \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$


Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form

:$\displaystyle \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$


Generalized Integration by Parts

Let $\map f x, \map g x$ be real functions which are integrable and at least $n$ times differentiable. Then: {{begin-eqn}} {{eqn | l = \int f^{\paren n} g \rd x       | r = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j f^{\paren {n - j - 1} } g^{\paren j} + \paren {-1}^n \int f g^{\paren n} \rd x       | c =  }} {{eqn | r = f^{\paren {n - 1} } g - f^{\paren {n - 2} } g' + f^{\paren {n - 3} } g'' - \cdots + \paren {-1}^n \int f g^{\paren n} \rd x       | c =  }} {{end-eqn}} where $f^{\paren n}$ denotes the $n$th derivative of $f$.


Primitive of Function of a x + b

:$\displaystyle \int \map F {a x + b} \rd x = \frac 1 a \int \map F u \rd u$ where $u = a x + b$.


Primitive of Function of Root of a x + b

:$\displaystyle \int F \left({\sqrt {a x + b}}\right) \ \mathrm d x = \frac 2 a \int u \ F \left({u}\right) \ \mathrm d u$ where $u = \sqrt {a x + b}$.


Derivative of Nth Root

Let $n \in \N_{>0}$. Let $f: \R \to \R$ be the real function defined as $f \left({x}\right) = \sqrt [n] x$. Then: :$f' \left({x}\right) = \dfrac 1 {n \left({\sqrt [n] x}\right)^{n-1} }$ everywhere that $f \left({x}\right) = \sqrt [n] x$ is defined.


Primitive of Function of Root of a squared minus x squared

:$\displaystyle \int F \left({\sqrt {a^2 - x^2}}\right) \ \mathrm d x = a \int \cos u \ F \left({a \cos u}\right) \ \mathrm d u$ where $x = a \sin u$.


Primitive of Function of Root of x squared minus a squared

:$\displaystyle \int F \left({\sqrt {x^2 - a^2}}\right) \ \mathrm d x = a \int \sec u \tan u \ F \left({a \tan u}\right) \ \mathrm d u$ where $x = a \sec u$.


Primitive of Function of Exponential Function

:$\displaystyle \int \map F {e^{a x} } \rd x = \frac 1 a \int \frac {\map F u} u \rd u$ where $u = e^{a x}$.


Primitive of Function of Arcsine

:$\displaystyle \int F \left({\arcsin \frac x a}\right) \ \mathrm d x = a \int F \left({u}\right) \cos u \ \mathrm d u$ where $u = \arcsin \dfrac x a$.


Primitive of Function of Arccosine

:$\displaystyle \int F \paren {\arccos \frac x a} \rd x = -a \int F \paren u \sin u \rd u$ where $u = \arccos \dfrac x a$.


Primitive of Function of Arctangent

:$\displaystyle \int F \left({\arctan \frac x a}\right) \ \mathrm d x = a \int F \left({u}\right) \sec^2 u \ \mathrm d u$ where $u = \arctan \dfrac x a$.


Primitive of Function of Arccotangent

:$\displaystyle \int F \left({\operatorname{arccot} \frac x a}\right) \ \mathrm d x = -a \int F \left({u}\right) \csc^2 u \ \mathrm d u$ where $u = \operatorname{arccot} \dfrac x a$.


Primitive of Function of Arcsecant

:$\displaystyle \int \map F {\arcsec \frac x a} \rd x = a \int \map F u \sec u \tan u \rd u$ where $u = \arcsec \dfrac x a$.


Primitive of Reciprocal of a x + b

:$\displaystyle \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$


Primitive of x over a x + b

:$\displaystyle \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$


Laplace Transform of Exponential times Function

:$\laptrans {e^{a t} \map f t} = \map F {s - a}$ everywhere that $\laptrans f$ exists, for $\map \Re s > a$


Laplace Transform of Derivative

Let $f'$ be piecewise continuous with one-sided limits on said intervals. Then $\laptrans f$ exists for $\map \Re s > a$, and: :$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0$


Primitive of x squared over a x + b

:$\displaystyle \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$


Primitive of x cubed over a x + b

:$\displaystyle \int \frac {x^3 \rd x} {a x + b} = \frac {\paren {a x + b}^3} {3 a^4} - \frac {3 b \paren {a x + b}^2} {2 a^4} - \frac {3 b^2 \paren {a x + b} } {a^4} + \frac {b^3} {a^4} \ln \size {a x + b} + C$


Primitive of Reciprocal of x by a x + b

:$\displaystyle \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$


Primitive of Reciprocal of x squared by a x + b

:$\ds \int \frac {\d x} {x^2 \paren {a x + b} } = -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$


Primitive of Reciprocal of x cubed by a x + b

:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({a x + b}\right)} = \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C$


Primitives of Rational Functions involving a x + b

This page gathers together the primitives of some rational functions involving $a x + b$.


Primitive of Reciprocal of a x + b squared

:$\displaystyle \int \frac {\d x} {\paren {a x + b}^2} = -\frac 1 {a \paren {a x + b} } + C$


Primitive of x over a x + b squared

:$\displaystyle \int \frac {x \ \mathrm d x} {\left({a x + b}\right)^2} = \frac b {a^2 \left({a x + b}\right)} + \frac 1 {a^2} \ln \left\vert{a x + b}\right\vert + C$


Primitive of x squared over a x + b squared

:$\displaystyle \int \frac {x^2 \rd x} {\paren {a x + b}^2} = \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C$


Primitive of x cubed over a x + b squared

:$\displaystyle \int \frac {x^3 \ \mathrm d x} {\left({a x + b}\right)^2} = \frac {\left({a x + b}\right)^2} {2 a^4} - \frac {3 b \left({a x + b}\right)} {a^4} + \frac {b^3} {a^4 \left({a x + b}\right)} + \frac {3 b^2} {a^4} \ln \left\vert{a x + b}\right\vert + C$


Primitive of Reciprocal of x by a x + b squared

:$\displaystyle \int \frac {\d x} {x \paren {a x + b}^2} = \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C$


Primitive of Reciprocal of x squared by a x + b squared

:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({a x + b}\right)^2} = \frac {-a} {b^2 \left({a x + b}\right)} - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \left\vert{\frac {a x + b} x}\right\vert + C$


Primitive of Reciprocal of x cubed by a x + b squared

:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({a x + b}\right)^2} = - \frac {\left({a x + b}\right)^2} {2 b^4 x^2} + \frac {3 a \left({a x + b}\right)} {b^4 x} - \frac {a^3 x} {b^4 \left({a x + b}\right)} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + C$


Primitives of Rational Functions involving a x + b squared

This page gathers together the primitives of some rational functions involving $\left({a x + b}\right)^2$.


Primitive of Reciprocal of a x + b cubed

:$\displaystyle \int \frac {\d x} {\paren {a x + b}^3} = -\frac 1 {2 a \paren {a x + b}^2} + C$


Primitive of x over a x + b cubed

:$\displaystyle \int \frac {x \ \mathrm d x} {\left({a x + b}\right)^3} = \frac {-1} {a^2 \left({a x + b}\right)} + \frac b {2 a^2 \left({a x + b}\right)^2} + C$


Primitive of Reciprocal of x by a x + b cubed

:$\displaystyle \int \frac {\mathrm d x} {x \left({a x + b}\right)^3} = \frac {a^2 x^2} {2 b^3 \left({a x + b}\right)^2} - \frac {2 a x} {b^3 \left({a x + b}\right)} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C$


Primitive of Reciprocal of x squared by a x + b cubed

:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({a x + b}\right)^3} = \frac {-a} {2 b^2 \left({a x + b}\right)^2} - \frac {2 a} {b^3 \left({a x + b}\right)} - \frac 1 {b^3 x} + \frac {3 a} {b^4} \ln \left\vert{\frac {a x + b} x}\right\vert + C$


Primitive of Reciprocal of x cubed by a x + b cubed

:$\displaystyle \int \frac {\d x} {x^3 \paren {a x + b}^3} = \frac {a^4 x^2} {2 b^5 \paren {a x + b}^2} - \frac {4 a^3 x} {b^5 \paren {a x + b} } - \frac {\paren {a x + b}^2} {2 b^5 x^2} + \frac {4 a} {b^4 x} + \frac {6 a^2} {b^5} \ln \size {\frac x {a x + b} } + C$


Laplace Transform of Second Derivative

Let $f: \R \to \R$ or $\R \to \C$ be a continuous function on any interval of the form $0 \le t \le a$. Let $f$ be twice differentiable. Let $f'$ be continuous and $f''$ piecewise continuous with one-sided limits on said intervals. Let $f$ and $f'$ be of exponential order. Let $\laptrans f$ denote the Laplace transform of $f$. Then $\laptrans {f''}$ exists for $\map \Re s > a$, and: :$\laptrans {\map {f''} t} = s^2 \laptrans {\map f t} - s \, \map f 0 - \map {f'} 0$


Euler's Number is Transcendental

Euler's Number $e$ is transcendental.


Laplace Transform of Hyperbolic Cosine

Let $\cosh t$ be the hyperbolic cosine, where $t$ is real. Let $\laptrans f$ denote the Laplace transform of the real function $f$. Then: :$\laptrans {\cosh a t} = \dfrac s {s^2 - a^2}$ where $a \in \R_{>0}$ is constant, and $\map \Re s > a$.


Laplace Transform of Hyperbolic Sine

Let $\sinh t$ be the hyperbolic sine, where $t$ is real. Let $\laptrans f$ denote the Laplace transform of the real function $f$. Then: :$\laptrans {\sinh a t} = \dfrac a {s^2 - a^2}$ where $a \in \R_{>0}$ is constant, and $\map \Re s > a$.


Primitives of Rational Functions involving a x + b cubed

This page gathers together the primitives of some rational functions involving $\left({a x + b}\right)^3$.


Primitive of Power of a x + b

:$\displaystyle \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$


Primitive of x by Power of a x + b

:$\displaystyle \int x \left({a x + b}\right)^n \ \mathrm d x = \frac {\left({a x + b}\right)^{n + 2} } {\left({n + 2}\right) a^2} - \frac {b \left({a x + b}\right)^{n + 1} } {\left({n + 1}\right) a^2} + C$


Primitive of x squared by Power of a x + b

:$\displaystyle \int x^2 \left({a x + b}\right)^n \ \mathrm d x = \frac {\left({a x + b}\right)^{n + 3} } {\left({n + 3}\right) a^3} - \frac {2 b \left({a x + b}\right)^{n + 2} } {\left({n + 2}\right) a^3} + \frac {b^2 \left({a x + b}\right)^{n + 1} } {\left({n + 1}\right) a^3} + C$


Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b

:$\displaystyle \int x^m \left({a x + b}\right)^n \rd x = \frac {x^{m+1} \left({a x + b}\right)^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \left({a x + b}\right)^{n - 1} \rd x$


Primitive of x over a x + b cubed/Proof 1


Primitive of x over a x + b cubed/Proof 2


=== Decrement of Power of $a x + b$ === {{:Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b}} === Decrement of Power of $x$ === {{:Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of x}} === Increment of Power of $a x + b$ === {{:Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of a x + b}} === Increment of Power of $x$ === {{:Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of x}}


Distance from Subset to Element

Let $\struct {M, d}$ be a metric space. Let $S \subseteq M$ be a subset of $M$. Let $s \in S$. Then: :$\map d {s, S} = 0$ where $\map d {s, S}$ denotes the distance between $s$ and $S$.


Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of x

:$\displaystyle \int x^m \left({a x + b}\right)^n \rd x = \frac {x^m \left({a x + b}\right)^{n + 1} } {\left({m + n + 1}\right) a} - \frac {m b} {\left({m + n + 1}\right) a} \int x^{m - 1} \left({a x + b}\right)^n \rd x$


Distance from Subset to Supremum

Let $S \subseteq \R$ be a subset of the real numbers. Suppose that the supremum $\sup S$ of $S$ exists. Then: :$\map d {\sup S, S} = 0$ where $\map d {\sup S, S}$ is the distance between $\sup S$ and $S$.


Distance between Element and Subset is Nonnegative

Let $\struct {M, d}$ be a metric space. Let $x \in M$ and $S \subseteq M$. Then: :$\map d {x, S} \ge 0$ where $\map d {x, S}$ is the distance between $x$ and $S$.


Distance from Subset to Infimum

Let $S \subseteq \R$ be a subset of the real numbers. Suppose that the infimum $\inf S$ of $S$ exists. Then: :$\map d {\inf S, S} = 0$ where $\map d {\inf S, S}$ is the distance between $\inf S$ and $S$.


Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of a x + b

:$\displaystyle \int x^m \paren {a x + b}^n \rd x = \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x$


Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of x

:$\displaystyle \int x^m \left({a x + b}\right)^n \rd x = \frac {x^{m+1} \left({a x + b}\right)^{n + 1} } {\left({m + 1}\right) b} - \frac {\left({m + n + 2}\right) a} {\left({m + 1}\right) b} \int x^{m + 1} \left({a x + b}\right)^n \rd x$


Primitive of Reciprocal of Root of a x + b

:$\displaystyle \int \frac {\d x} {\sqrt{a x + b} } = \frac {2 \sqrt {a x + b} } a + C$


Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form

:$\ds \int \frac {\d x} {x^2 - a^2} = -\frac 1 a \coth^{-1} {\frac x a} + C$ where $\size x > a$.


Primitive of Reciprocal of x by Root of a x + b

:$\displaystyle \int \frac {\d x} {x \sqrt{a x + b} } = \begin {cases} \dfrac 1 {\sqrt b} \map \ln {\dfrac {\sqrt {a x + b} - \sqrt b} {\sqrt {a x + b} + \sqrt b} } + C & : b > 0 \\ \dfrac 2 {\sqrt {-b} } \arctan \sqrt {\dfrac {a x + b} {-b} } + C & : b < 0 \end {cases}$


Primitive of Reciprocal of x squared by Root of a x + b

:$\displaystyle \int \frac {\d x} {x^2 \sqrt{a x + b} } = -\frac {\sqrt{a x + b} } {b x} - \frac a {2 b} \int \frac {\mathrm d x} {x \sqrt{a x + b} }$


Primitive of Root of a x + b

:$\displaystyle \int \sqrt{a x + b} \ \mathrm d x = \frac {2 \sqrt{\left({a x + b}\right)^3} } {3 a}$


Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form

:$\ds \int \frac {\d x} {a^2 - x^2} = \frac 1 a \tanh^{-1} \frac x a + C$ where $\size x < a$.


Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form

:$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \sinh^{-1} {\frac x a} + C$


Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form

:$\displaystyle \int \frac {\d x} {\sqrt {x^2 - a^2} } = \dfrac {\size x} x \cosh^{-1} {\size {\frac x a} } + C$ for $x > a$.


Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form

:$\displaystyle \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \csch^{-1} {\frac x a} + C$


Primitive of Reciprocal of x by Root of a squared minus x squared/Inverse Hyperbolic Secant Form

:$\displaystyle \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \sech^{-1} {\frac x a} + C$


Primitive of x by Root of a x + b

:$\displaystyle \int x \sqrt{a x + b} \ \mathrm d x = \frac {2 \left({3 a x - 2 b}\right) } {15 a^2} \sqrt{\left({a x + b}\right)^3}$


Laplace Transform of Heaviside Step Function times Function

Let $\map f t: \R \to \R$ or $\R \to \C$ be a function of exponential order $a$ for some constant $a \in \R$. Let $f$ be piecewise continuous with one-sided limits on any closed interval of the form $\closedint 0 b$ where $b > 0$. Let $\map {u_c} t$ be the Heaviside step function. Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$. Then: :$\laptrans {\map {u_c} t \, \map f {t - c} } = e^{-s c} \map F s$ for $\map \Re s > a$.


Primitive of x squared by Root of a x + b

:$\displaystyle \int x^2 \sqrt{a x + b} \ \mathrm d x = \frac {2 \left({15 a^2 x^2 - 12 a b x + 8 b^2}\right) } {105 a^3} \sqrt{\left({a x + b}\right)^3} + C$


Primitive of Root of a x + b over x

:$\displaystyle \int \frac {\sqrt {a x + b} } x \rd x = 2 \sqrt {a x + b} + b \int \frac {\d x} {x \sqrt{a x + b} }$


Primitive of Root of a x + b over x squared

:$\displaystyle \int \frac {\sqrt {a x + b} } {x^2} \rd x = -\frac {\sqrt {a x + b} } x + \frac a 2 \int \frac {\d x} {x \sqrt {a x + b} }$


Primitive of Power of x over Root of a x + b

:$\ds \int \frac {x^m} {\sqrt{a x + b} } \rd x = \frac {2 x^m \sqrt{a x + b} } {\paren {2 m + 1} a} - \frac {2 m b} {\paren {2 m + 1} a} \int \frac {x^{m - 1} } {\sqrt{a x + b} } \rd x$


Primitive of Reciprocal of Power of x by Root of a x + b

:$\displaystyle \int \frac {\d x} {x^m \sqrt {a x + b} } = -\frac {\sqrt {a x + b} } {\paren {m - 1} b x^{m - 1} } - \frac {\paren {2 m - 3} a} {\paren {2 m - 2} b} \int \frac {\d x} {x^{m - 1} \sqrt {a x + b} }$


=== Formulation 1 === {{:Primitive of Root of a x + b over Power of x/Formulation 1}} === Formulation 2 === {{:Primitive of Root of a x + b over Power of x/Formulation 2}}


Primitive of Power of Root of a x + b over x

:$\displaystyle \int \frac {\left({\sqrt{a x + b} }\right)^m} x \ \mathrm d x = \frac {2 \left({\sqrt{a x + b} }\right)^m } m + b \int \frac {\left({\sqrt{a x + b} }\right)^{m - 2} } x \ \mathrm d x$


Primitive of Reciprocal of x by Power of Root of a x + b

:$\displaystyle \int \frac {\mathrm d x} {x \left({\sqrt{a x + b} }\right)^m} = \frac 2 {\left({m - 2}\right) b \left({\sqrt{a x + b} }\right)^{m - 2} }  + \frac 1 b \int \frac {\mathrm d x} {x \left({\sqrt{a x + b} }\right)^{m - 2} }$


Primitive of Reciprocal of a x + b by p x + q

:$\displaystyle \int \frac {\mathrm d x} {\left({a x + b}\right) \left({p x + q}\right)} = \frac 1 {b p - a q} \ln \left\vert{\frac {p x + q} {a x + b} }\right\vert + C$


Primitive of x over a x + b by p x + q

:$\displaystyle \int \frac {x \ \mathrm d x} {\left({a x + b}\right) \left({p x + q}\right)} = \frac 1 {b p - a q} \left({\frac b a \ln \left\vert{a x + b}\right\vert - \frac q p \ln \left\vert{p x + q}\right\vert}\right) + C$


Primitive of Reciprocal of a x + b squared by p x + q

:$\displaystyle \int \frac {\mathrm d x} {\left({a x + b}\right)^2 \left({p x + q}\right)} = \frac 1 {b p - a q} \left({\frac 1 {a x + b} + \frac p {b p - a q} \ln \left\vert{\frac {p x + q} {a x + b} }\right\vert}\right) + C$


Primitive of x over a x + b squared by p x + q

:$\displaystyle \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } - \frac b {a \paren {a x + b} } } + C$


Primitive of x squared over a x + b squared by p x + q

:$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({a x + b}\right)^2 \left({p x + q}\right)} = \frac {b^2} {\left({b p - a q}\right) a^2 \left({a x + b}\right)} + \frac 1 {\left({b p - a q}\right)^2} \left({\frac {q^2} p \ln \left\vert{p x + q}\right\vert + \frac {b \left({b p - 2 a q}\right)} {a^2} \ln \left\vert{a x + b}\right\vert}\right) + C$


Friedrichs' Inequality

Let $G \subset \R^n$ be bounded domain. Then for any $u \in \map {W^{2, 1}_0} G$: :$\norm u_{\map {L^2} G} \le \map {\operatorname {diam} } G \norm {\nabla u}_{\map {L^2} G}$ where: :$\map {\operatorname {diam} } G := \sup \limits_{x, y \mathop \in G} \size {x - y}$


Primitive of Reciprocal of Power of a x + b by Power of p x + q

:$\displaystyle \int \frac {\mathrm d x} {\left({a x + b}\right)^m \left({p x + q}\right)^n} = \frac {-1} {\left({n - 1}\right) \left({b p - a q}\right)} \left({\frac 1 {\left({a x + b}\right)^{m-1} \left({p x + q}\right)^{n-1} } + a \left({m + n - 2}\right) \int \frac {\mathrm d x} {\left({a x + b}\right)^m \left({p x + q}\right)^{n-1} } }\right)$


Reduction Formula for Primitive of Power of a x + b by Power of p x + q/Decrement of Power

:$\displaystyle \int \left({a x + b}\right)^m \left({p x + q}\right)^n \ \mathrm d x = \frac {\left({a x + b}\right)^{m+1} \left({p x + q}\right)^n} {\left({m + n + 1}\right) a} - \frac {n \left({b p - a q}\right)} {\left({m + n + 1}\right) a} \int \left({a x + b}\right)^m \left({p x + q}\right)^{n-1} \ \mathrm d x$


Reduction Formula for Primitive of Power of a x + b by Power of p x + q/Increment of Power

:$\displaystyle \int \left({a x + b}\right)^m \left({p x + q}\right)^n \ \mathrm d x = \frac 1 {\left({n + 1}\right) \left({b p - a q}\right)} \left({\left({a x + b}\right)^{m+1} \left({p x + q}\right)^{n+1} - a \left({m + n + 2}\right) \int \left({a x + b}\right)^m \left({p x + q}\right)^{n+1} \ \mathrm d x}\right)$


Primitive of a x + b over p x + q

:$\displaystyle \int \frac {a x + b} {p x + q} \ \mathrm d x  = \frac {a x} p + \frac {b p - a q} {p^2} \ln \left\vert{p x + q}\right\vert + C$


Reduction Formula for Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b/Proof 1

:$\displaystyle \int x^m \left({a x + b}\right)^n \rd x = \frac {x^{m+1} \left({a x + b}\right)^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \left({a x + b}\right)^{n - 1} \rd x$


Primitive of Power of a x + b over Power of p x + q/Formulation 1

:$\displaystyle \int \frac {\left({a x + b}\right)^m} {\left({p x + q}\right)^n} \rd x = \frac {-1} {\left({n - 1}\right) \left({b p - a q}\right)} \left({\frac {\left({a x + b}\right)^{m + 1} } {\left({p x + q}\right)^{n - 1} } + \left({n - m - 2}\right) a \int \frac {\left({a x + b}\right)^m} { \left({p x + q}\right)^{n - 1} } \rd x}\right)$


=== Formulation 1 === {{:Primitive of Power of a x + b over Power of p x + q/Formulation 1}} === Formulation 2 === {{:Primitive of Power of a x + b over Power of p x + q/Formulation 2}} === Formulation 3 === {{:Primitive of Power of a x + b over Power of p x + q/Formulation 3}}


Primitive of Power of a x + b over Power of p x + q/Formulation 2

:$\displaystyle \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - m - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } + m \paren {b p - a q} \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}^n} \rd x}$


Primitive of Power of a x + b over Power of p x + q/Formulation 3

:$\displaystyle \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } - m a \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}^{n - 1}} \rd x}$


Primitive of Reciprocal of p x + q by Root of a x + b

:$\displaystyle \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} } = \begin {cases} \dfrac 1 {\sqrt {b p - a q} \sqrt p} \ln \size {\dfrac {\sqrt {p \paren {a x + b} } - \sqrt {b p - a q} } {\sqrt {p \paren {a x + b} } + \sqrt {b p - a q} } } & : b p - a q > 0 \\ \dfrac 2 {\sqrt {a q - b p} \sqrt p} \arctan \sqrt {\dfrac {p \paren {a x + b} } {a q - b p} } & : b p - a q < 0 \\ \end {cases}$


Primitive of Root of a x + b over p x + q

:$\displaystyle \int \frac {\sqrt{a x + b} } {p x + q} \rd x = \begin{cases} \dfrac {2 \sqrt{a x + b} } p + \dfrac {\sqrt {b p - a q} } {p \sqrt p} \ln \size {\dfrac {\sqrt {p \paren {a x + b} } - \sqrt {b p - a q} } {\sqrt {p \paren {a x + b} } + \sqrt {b p - a q} } } & : b p - a q > 0 \\ \dfrac {2 \sqrt{a x + b} } p - \dfrac {\sqrt {a q - b p} } {p \sqrt p} \arctan \sqrt {\dfrac {p \paren {a x + b} } {a q - b p} } & : b p - a q < 0 \\ \end{cases}$


Primitive of Power of p x + q by Root of a x + b

:$\displaystyle \int \paren {p x + q}^n \sqrt {a x + b} \rd x = \frac {2 \paren {p x + q}^{n + 1} \sqrt {a x + b} } {\paren {2 n + 3} p} + \frac {b p - a q} {\paren {2 n + 3} p} \int \frac {\paren {p x + q}^n} {\sqrt{a x + b} } \rd x$


Primitive of Reciprocal of Power of p x + q by Root of a x + b

:$\displaystyle \int \frac {\mathrm d x} {\left({p x + q}\right)^n \sqrt{a x + b} } = \frac {\sqrt{a x + b} } {\left({n - 1}\right) \left({a q - b p}\right) \left({p x + q}\right)^{n-1} } + \frac {\left({2 n - 3}\right) a} {2 \left({n - 1}\right)  \left({a q - b p}\right)} \int \frac {\mathrm d x} {\left({p x + q}\right)^{n-1} \sqrt{a x + b} }$


Primitive of Power of p x + q over Root of a x + b

:$\displaystyle \int \frac {\left({p x + q}\right)^n} {\sqrt{a x + b} } \ \mathrm d x = \frac {2 \left({p x + q}\right)^n \sqrt{a x + b} } {\left({2 n + 1}\right) a} + \frac {2 n \left({a q - b p}\right)} {\left({2 n + 1}\right) a} \int \frac {\left({p x + q}\right)^{n-1} } {\sqrt{a x + b} } \ \mathrm d x$


Primitive of Reciprocal of Root of a x + b by Root of p x + q

:$\displaystyle \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \begin {cases} \dfrac 2 {\sqrt {a p} } \map \ln {\sqrt {p \paren {a x + b} } + \sqrt {a \paren {p x + q} } } + C & : \dfrac {b p - a q} p > 0 \\ \dfrac 2 {\sqrt {a p} } \sinh^{-1} \sqrt {\dfrac {p \paren {a x + b} } {b p - a q} } + C & : \dfrac {b p - a q} p < 0 \\ \end {cases}$


Derivative of Laplace Transform

Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, differentiable on any closed interval $\closedint 0 a$. Let $\laptrans f = F$ denote the Laplace transform of $f$. Then, everywhere that $\dfrac \d {\d s} \laptrans f$ exists: :$\dfrac \d {\d s} \laptrans {\map f t} = -\laptrans {t \, \map f t}$


Primitive of x over Root of a x + b by Root of p x + q

:$\displaystyle \int \frac {x \ \mathrm d x} {\sqrt{\left({a x + b}\right) \left({p x + q}\right)} } = \frac {\sqrt{\left({a x + b}\right) \left({p x + q}\right)} } {a p} - \frac {b p + a q} {2 a p} \int \frac {\mathrm d x} {\sqrt{\left({a x + b}\right) \left({p x + q}\right)} }$


Higher Order Derivatives of Laplace Transform

:$\dfrac {\d^n} {\d s^n} \laptrans {\map f t} = \paren {-1}^n \laptrans {t^n \, \map f t}$


Primitive of Root of a x + b by Root of p x + q

:$\displaystyle \int \sqrt {\paren {a x + b} \paren {p x + q} } \rd x = \frac {2 a p x + b p + a q} {4 a p} \sqrt {\paren {a x + b} \paren {p x + q} } - \frac {\paren {b p - a q}^2} {8 a p} \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }$


Primitive of Root of p x + q over Root of a x + b

:$\displaystyle \int \sqrt {\frac {p x + q} {a x + b} } \rd x = \frac {\sqrt {\paren {a x + b} \paren {p x + q} } } a + \frac {a q - b p} {2 a} \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }$


Primitive of Reciprocal of p x + q by Root of a x + b by Root of p x + q

:$\displaystyle \int \frac {\d x} {\paren {p x + q} \sqrt {\paren {a x + b} \paren {p x + q} } } = \frac {2 \sqrt{a x + b} } {\paren {a q - b p} \sqrt {p x + q} } + C$


Primitive of x over x squared plus a squared

:$\displaystyle \int \frac {x \rd x} {x^2 + a^2} = \frac 1 2 \map \ln {x^2 + a^2} + C$


Primitive of x squared over x squared plus a squared

:$\displaystyle \int \frac {x^2 \rd x} {x^2 + a^2} = x - a \arctan {\frac x a} + C$


Primitive of x cubed over x squared plus a squared

:$\displaystyle \int \frac {x^3 \rd x} {x^2 + a^2} = \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$


Primitive of Reciprocal of x by x squared plus a squared

:$\ds \int \frac {\rd x} {x \paren {x^2 + a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 + a^2} } + C$


Primitive of Reciprocal of x squared by x squared plus a squared

:$\displaystyle \int \frac {\d x} {x^2 \paren {x^2 + a^2} } = -\frac 1 {a^2 x} - \frac 1 {a^3} \arctan \frac x a + C$


Primitive of Reciprocal of x cubed by x squared plus a squared

:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({x^2 + a^2}\right)} = -\frac 1 {2 a^2 x^2} - \frac 1 {2 a^4} \ln \left({\frac {x^2 + a^2} {x^2} }\right) + C$


Primitive of Reciprocal of x squared plus a squared squared

:$\displaystyle \int \frac {\d x} {\paren {x^2 + a^2}^2} = \frac x {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^3} \arctan \frac x a + C$


Primitive of x over x squared plus a squared squared

:$\displaystyle \int \frac {x \ \mathrm d x} {\left({x^2 + a^2}\right)^2} = -\frac 1 {2 \left({x^2 + a^2}\right)} + C$


Primitive of x squared over x squared plus a squared squared

:$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({x^2 + a^2}\right)^2} = \frac {-x} {2 \left({x^2 + a^2}\right)} + \frac 1 {2 a} \arctan \frac x a + C$


Primitive of x cubed over x squared plus a squared squared

:$\displaystyle \int \frac {x^3 \ \mathrm d x} {\left({x^2 + a^2}\right)^2} = \frac {a^2} {2 \left({x^2 + a^2}\right)} + \frac 1 2 \ln \left({x^2 + a^2}\right) + C$


Primitive of Reciprocal of x by x squared plus a squared squared

:$\displaystyle \int \frac {\d x} {x \paren {x^2 + a^2}^2} = \frac 1 {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {x^2 + a^2} } + C$


Primitive of Reciprocal of x squared by x squared plus a squared squared

:$\displaystyle \int \frac {\d x} {x^2 \paren {x^2 + a^2}^2} = -\frac 1 {a^4 x} - \frac x {2 a^4 \paren {x^2 + a^2} } - \frac 3 {2 a^5} \arctan \frac x a + C$


Primitive of Reciprocal of x cubed by x squared plus a squared squared

:$\displaystyle \int \frac {\d x} {x^3 \paren {x^2 + a^2}^2} = -\frac 1 {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 + a^2} } - \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 + a^2} } + C$


Primitive of Reciprocal of Power of x squared plus a squared

:$\displaystyle \int \frac {\mathrm d x} {\left({x^2 + a^2}\right)^n} = \frac x {2 \left({n - 1}\right) a^2 \left({x^2 + a^2}\right)^{n - 1} } + \frac {2 n - 3} {\left({2 n - 2}\right) a^2} \int \frac {\mathrm d x} {\left({x^2 + a^2}\right)^{n - 1} }$


Primitive of Reciprocal of x by Power of x squared plus a squared

:$\displaystyle \int \frac {\mathrm d x} {x \left({x^2 + a^2}\right)^n} = \frac 1 {2 \left({n - 1}\right) a^2 \left({x^2 + a^2}\right)^{n - 1} } + \frac 1 {a^2} \int \frac {\mathrm d x} {x \left({x^2 + a^2}\right)^{n - 1} }$


Primitive of Reciprocal of Power of x by Power of x squared plus a squared

:$\displaystyle \int \frac {\d x} {x^m \paren {x^2 + a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 + a^2}^{n - 1} } - \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 + a^2}^n}$


Primitive of x over x squared minus a squared

:$\displaystyle \int \frac {x \rd x} {x^2 - a^2} = \frac 1 2 \map \ln {x^2 - a^2} + C$ for $x^2 > a^2$.


Primitive of x cubed over x squared minus a squared

:$\displaystyle \int \frac {x^3 \ \mathrm d x} {x^2 - a^2} = \frac {x^2} 2 + \frac {a^2} 2 \ln \left({x^2 - a^2}\right) + C$ for $x^2 > a^2$.


Primitive of Reciprocal of x by x squared minus a squared

:$\displaystyle \int \frac {\d x} {x \paren {x^2 - a^2} } = \frac 1 {2 a^2} \, \map \ln {\frac {x^2 - a^2} {x^2} } + C$ for $x^2 > a^2$.


Primitive of Reciprocal of x squared by x squared minus a squared

:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^2 - a^2}\right)} = \frac 1 {a^2 x} + \frac 1 {2 a^3} \ln \left({\frac {x - a} {x + a} }\right) + C$ for $x^2 > a^2$.


Primitive of Reciprocal of x cubed by x squared minus a squared

:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({x^2 - a^2}\right)} = \frac 1 {2 a^2 x^2} + \frac 1 {2 a^4} \ln \left({\frac {x^2 - a^2} {x^2} }\right) + C$ for $x^2 > a^2$.


Primitive of Reciprocal of x squared minus a squared squared

:$\displaystyle \int \frac {\d x} {\left({x^2 - a^2}\right)^2} = \frac {-x} {2 a^2 \left({x^2 - a^2}\right)} + \frac 1 {4 a^3} \ln \left({\frac {x + a} {x - a} }\right) + C$ for $x^2 > a^2$.


Primitive of x over x squared minus a squared squared

:$\displaystyle \int \frac {x \ \mathrm d x} {\left({x^2 - a^2}\right)^2} = \frac {-1} {2 \left({x^2 - a^2}\right)} + C$ for $x^2 > a^2$.


Primitive of x squared over x squared minus a squared squared

:$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({x^2 - a^2}\right)^2} = \frac {-x} {2 \left({x^2 - a^2}\right)} + \frac 1 {4 a} \ln \left({\frac {x - a} {x + a} }\right) + C$ for $x^2 > a^2$.


Primitive of x cubed over x squared minus a squared squared

:$\displaystyle \int \frac {x^3 \ \mathrm d x} {\left({x^2 - a^2}\right)^2} = \frac {-a^2} {2 \left({x^2 - a^2}\right)} + \frac 1 2 \ln \left({x^2 - a^2}\right) + C$ for $x^2 > a^2$.


Primitive of Reciprocal of x by x squared minus a squared squared

:$\displaystyle \int \frac {\d x} {x \paren {x^2 - a^2}^2} = \frac {-1} {2 a^2 \left({x^2 - a^2}\right)} + \frac 1 {2 a^4} \ln \left({\frac {x^2} {x^2 - a^2} }\right) + C$ for $x^2 > a^2$.


Primitive of Reciprocal of x squared by x squared minus a squared squared

:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^2 - a^2}\right)^2} = \frac {-1} {a^4 x} - \frac x {2 a^4 \left({x^2 - a^2}\right)} + \frac 3 {4 a^5} \ln \left({\frac {x + a} {x - a} }\right) + C$ for $x^2 > a^2$.


Primitive of Reciprocal of x cubed by x squared minus a squared squared

:$\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2}^2} = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 - a^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 - a^2} } + C$ for $x^2 > a^2$.


Primitive of Reciprocal of Power of x squared minus a squared

:$\displaystyle \int \frac {\mathrm d x} {\left({x^2 - a^2}\right)^n} = \frac {-x} {2 \left({n - 1}\right) a^2 \left({x^2 - a^2}\right)^{n - 1} } - \frac {2 n - 3} {\left({2 n - 2}\right) a^2} \int \frac {\mathrm d x} {\left({x^2 - a^2}\right)^{n - 1} }$ for $x^2 > a^2$.


Primitive of x over Power of x squared minus a squared

:$\displaystyle \int \frac {x \rd x} {\paren {x^2 - a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 - a^2}^{n - 1} }$ for $x^2 > a^2$.


Primitive of Power of x over Power of x squared minus a squared

:$\displaystyle \int \frac {x^m \rd x} {\paren {x^2 - a^2}^n} = \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^{n - 1} } + a^2 \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^n}$ for $x^2 > a^2$.


Sign of Quotient of Factors of Difference of Squares

Let $a, b \in \R$ such that $a \ne b$. Then :$\map \sgn {a^2 - b^2} = \map \sgn {\dfrac {a + b} {a - b} } = \map \sgn {\dfrac {a - b} {a + b} }$ where $\sgn$ denotes the signum of a real number.


Signum Function is Completely Multiplicative

The signum function on the set of real numbers is a completely multiplicative function: :$\forall x, y \in \R: \map \sgn {x y} = \map \sgn x \map \sgn y$


Absolute Value Function is Completely Multiplicative

The absolute value function on the real numbers $\R$ is completely multiplicative: :$\forall x, y \in \R: \left\vert{x y}\right\vert = \left\vert{x}\right\vert \, \left\vert{y}\right\vert$ where $\left \vert{a}\right \vert$ denotes the absolute value of $a$.


Signum Function of Reciprocal

Let $x \in \R$ such that $x \ne 0$. Then: :$\map \sgn x = \map \sgn {\dfrac 1 x}$ where $\map \sgn x$ denotes the signum of $x$.


Primitive of x over a squared minus x squared

:$\displaystyle \int \frac {x \rd x} {a^2 - x^2} = -\frac 1 2 \, \map \ln {a^2 - x^2} + C$ for $x^2 < a^2$.


Linear Combination of Laplace Transforms

Then: :$\laptrans {\lambda \, \map f t + \mu \, \map g t} = \lambda \laptrans {\map f t} + \mu \laptrans {\map g t}$ everywhere all the above expressions are defined.


Primitive of x cubed over a squared minus x squared

:$\displaystyle \int \frac {x^3 \ \mathrm d x} {a^2 - x^2} = -\frac {x^2} 2 - \frac {a^2} 2 \ln \left({a^2 - x^2}\right) + C$ for $x^2 < a^2$.


Primitive of Reciprocal of x squared by a squared minus x squared

:$\displaystyle \int \frac {\d x} {x^2 \paren {a^2 - x^2} } = \frac {-1} {a^2 x} + \frac 1 {2 a^3} \map \ln {\frac {a + x} {a - x} } + C$ for $x^2 < a^2$.


Primitive of x over a squared minus x squared squared

:$\displaystyle \int \frac {x \ \mathrm d x} {\left({a^2 - x^2}\right)^2} = \frac 1 {2 \left({a^2 - x^2}\right)} + C$ for $x^2 < a^2$.


Primitive of x squared over a squared minus x squared squared

:$\displaystyle \int \frac {x^2 \rd x} {\paren {a^2 - x^2}^2} = \frac x {2 \paren {a^2 - x^2} } - \frac 1 {4 a} \map \ln {\frac {a + x} {a - x} } + C$ for $x^2 < a^2$.


Primitive of x cubed over a squared minus x squared squared

:$\displaystyle \int \frac {x^3 \rd x} {\paren {a^2 - x^2}^2} = \frac {a^2} {2 \paren {a^2 - x^2} } + \frac 1 2 \map \ln {a^2 - x^2} + C$ for $x^2 < a^2$.


Primitive of Reciprocal of x by a squared minus x squared squared

:$\displaystyle \int \frac {\d x} {x \paren {a^2 - x^2}^2} = \frac 1 {2 a^2 \paren {a^2 - x^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {a^2 - x^2} } + C$ for $x^2 < a^2$.


Primitive of Reciprocal of x squared by a squared minus x squared squared

:$\displaystyle \int \frac {\d x} {x^2 \paren {a^2 - x^2}^2} = \frac {-1} {a^4 x} + \frac x {2 a^4 \paren {a^2 - x^2} } + \frac 3 {4 a^5} \map \ln {\frac {a + x} {a - x} } + C$ for $x^2 < a^2$.


Primitive of Reciprocal of Power of a squared minus x squared

:$\displaystyle \int \frac {\mathrm d x} {\left({a^2 - x^2}\right)^n} = \frac x {2 \left({n - 1}\right) a^2 \left({a^2 - x^2}\right)^{n - 1} } + \frac {2 n - 3} {\left({2 n - 2}\right) a^2} \int \frac {\mathrm d x} {\left({a^2 - x^2}\right)^{n - 1} }$ for $x^2 > a^2$.


Primitive of Power of x over Power of a squared minus x squared

:$\displaystyle \int \frac {x^m \rd x} {\paren {a^2 - x^2}^n} = a^2 \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^n} - \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^{n - 1} }$ for $x^2 < a^2$.


Primitive of Reciprocal of Power of x by Power of a squared minus x squared

:$\displaystyle \int \frac {\mathrm d x} {x^m \left({a^2 - x^2}\right)^n} = \frac 1 {a^2} \int \frac {\mathrm d x} {x^m \left({a^2 - x^2}\right)^{n-1} } + \frac 1 {a^2} \int \frac {\mathrm d x} {x^{m-2} \left({a^2 - x^2}\right)^n}$ for $x^2 < a^2$.


Primitive of x over Root of x squared plus a squared

:$\ds \int \frac {x \rd x} {\sqrt {x^2 + a^2} } = \sqrt {x^2 + a^2} + C$


Primitive of x cubed over Root of x squared plus a squared

:$\displaystyle \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 - a^2 \sqrt {x^2 + a^2} + C$


Primitive of Reciprocal of x squared by Root of x squared plus a squared

:$\displaystyle \int \frac {\mathrm d x} {x^2 \sqrt {x^2 + a^2} } = \frac {-\sqrt {x^2 + a^2} } {a^2 x} + C$


Primitive of Reciprocal of x cubed by Root of x squared plus a squared

:$\displaystyle \int \frac {\mathrm d x} {x^3 \sqrt {x^2 + a^2} } = \frac {- \sqrt {x^2 + a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$


Primitive of x by Root of x squared plus a squared

:$\displaystyle \int x \sqrt {x^2 + a^2} \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + C$


Primitive of Root of x squared plus a squared over x

:$\displaystyle \int \frac {\sqrt {x^2 + a^2} } x \ \mathrm d x = \sqrt {x^2 + a^2} - a \ln \left({\frac {a + \sqrt {x^2 + a^2} } a}\right) + C$


Primitive of Root of x squared plus a squared over x squared

:$\displaystyle \int \frac {\sqrt {x^2 + a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 + a^2} } x + \map \ln {x + \sqrt {x^2 + a^2} } + C$


Primitive of Root of x squared plus a squared over x cubed

:$\displaystyle \int \frac {\sqrt {x^2 + a^2} } {x^3} \ \mathrm d x = \frac {-\sqrt {x^2 + a^2} } {2 x^2} - \frac 1 {2 a} \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$


Primitive of Reciprocal of Root of x squared plus a squared cubed

:$\ds \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac x {a^2 \sqrt {x^2 + a^2} } + C$


Primitive of x over Root of x squared plus a squared cubed

:$\displaystyle \int \frac {x \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac {-1} {\sqrt {x^2 + a^2} } + C$


Primitive of Reciprocal of x by Root of x squared plus a squared cubed

:$\displaystyle \int \frac {\d x} {x \paren {\sqrt {x^2 + a^2} }^3} = \frac 1 {a^2 \sqrt {x^2 + a^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$


Primitive of Reciprocal of x cubed by Root of x squared plus a squared cubed

:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({\sqrt {x^2 + a^2} }\right)^3} = \frac {-1} {2 a^2 x^2 \sqrt {x^2 + a^2} } - \frac 3 {2 a^4 \sqrt {x^2 + a^2} } + \frac 3 {2 a^5} \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$


Primitive of Root of x squared plus a squared cubed

:$\ds \int \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 + \frac {3 a^2 x \sqrt {x^2 + a^2} } 8 + \frac {3 a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$


Primitive of x by Root of x squared plus a squared cubed

:$\displaystyle \int x \left({\sqrt {x^2 + a^2} }\right)^3 \ \mathrm d x = \frac {\left({\sqrt {x^2 + a^2} }\right)^5} 5 + C$


Primitive of x squared by Root of x squared plus a squared cubed

:$\displaystyle \int x^2 \left({\sqrt {x^2 + a^2} }\right)^3 \ \mathrm d x = \frac {x \left({\sqrt {x^2 + a^2} }\right)^5} 6 - \frac {a^2 x \left({\sqrt {x^2 + a^2} }\right)^3} {24} - \frac {a^4 x \sqrt {x^2 + a^2} } {16} - \frac {a^6} {16} \ln \left({x + \sqrt {x^2 + a^2} }\right) + C$


Primitive of x cubed by Root of x squared plus a squared cubed

:$\displaystyle \int x^3 \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^7} 7 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^5} 5 + C$


Primitive of x over Root of x squared minus a squared

:$\displaystyle \int \frac {x \rd x} {\sqrt {x^2 - a^2} } = \sqrt {x^2 - a^2} + C$


{{:Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form}}


Primitive of x cubed over Root of x squared minus a squared

:$\displaystyle \int \frac {x^3 \rd x} {\sqrt {x^2 - a^2} } = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + a^2 \sqrt {x^2 - a^2} + C$


Primitive of Reciprocal of x squared by Root of x squared minus a squared

:$\displaystyle \int \frac {\d x} {x^2 \sqrt {x^2 - a^2} } = \frac {\sqrt {x^2 - a^2} } {a^2 x} + C$


Primitive of Reciprocal of x cubed by Root of x squared minus a squared

:$\displaystyle \int \frac {\d x} {x^3 \sqrt {x^2 - a^2} } = \frac {\sqrt {x^2 - a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \arcsec \size {\frac x a} + C$


{{:Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form}}


Primitive of x by Root of x squared minus a squared

:$\displaystyle \int x \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + C$


Primitive of x squared by Root of x squared minus a squared

:$\displaystyle \int x^2 \sqrt {x^2 - a^2} \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 + \frac {a^2 x \sqrt {x^2 - a^2} } 8 - \frac {a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$


Primitive of Root of x squared minus a squared over x

:$\displaystyle \int \frac {\sqrt {x^2 - a^2} } x \rd x = \sqrt {x^2 - a^2} - \frac 1 {2 a} \arcsec \size {\frac x a} + C$


Primitive of Root of x squared minus a squared over x cubed

:$\displaystyle \int \frac {\sqrt {x^2 - a^2} } {x^3} \rd x = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 {2 a} \arcsec \size {\frac x a} + C$


Primitive of Reciprocal of Root of x squared minus a squared cubed

:$\displaystyle \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {a^2 \sqrt {x^2 - a^2} } + C$


Primitive of x over Root of x squared minus a squared cubed

:$\displaystyle \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-1} {\sqrt {x^2 - a^2} } + C$


Primitive of x squared over Root of x squared minus a squared cubed

:$\displaystyle \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {\sqrt {x^2 - a^2} } + \ln \size {x + \sqrt {x^2 - a^2} } + C$


Primitive of x cubed over Root of x squared minus a squared cubed

:$\displaystyle \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \sqrt {x^2 - a^2} - \frac {a^2} {\sqrt {x^2 - a^2} } + C$


Primitive of Reciprocal of x by Root of x squared minus a squared cubed

:$\displaystyle \int \frac {\d x} {x \paren {\sqrt {x^2 - a^2} }^3} = \frac {-1} {a^2 \sqrt {x^2 - a^2} } - \frac 1 {a^3} \arcsec \size {\frac x a} + C$


Primitive of Reciprocal of x cubed by Root of x squared minus a squared cubed

:$\displaystyle \int \frac {\d x} {x^3 \paren {\sqrt {x^2 - a^2} }^3} = \frac 1 {2 a^2 x^2 \sqrt {x^2 - a^2} } - \frac 3 {2 a^4 \sqrt {x^2 - a^2} } + \frac 3 {2 a^5} \arcsec \size {\frac x a} + C$


Primitive of Root of x squared minus a squared cubed

:$\displaystyle \int \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 - \frac {3 a^2 x \sqrt {x^2 - a^2} } 8 + \frac {3 a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$


Primitive of x by Root of x squared minus a squared cubed

:$\displaystyle \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + C$


Primitive of x cubed by Root of x squared minus a squared cubed

:$\displaystyle \int x^3 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^7} 7 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^5} 5 + C$


Primitive of Root of x squared minus a squared cubed over x squared

:$\displaystyle \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {x^2 - a^2} }^3} x + \frac{3 x \sqrt {x^2 - a^2} } 2 - \frac {3 a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$


Primitive of Root of x squared minus a squared cubed over x cubed

:$\displaystyle \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {x^2 - a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 - a^2} } 2 - \frac {3 a} 2 \arcsec \size {\frac x a} + C$


Primitive of x over Root of a squared minus x squared

:$\displaystyle \int \frac {x \rd x} {\sqrt {a^2 - x^2} } = -\sqrt {a^2 - x^2} + C$


Primitive of x cubed over Root of a squared minus x squared

:$\displaystyle \int \frac {x^3 \rd x} {\sqrt {a^2 - x^2} } = \frac {\left({\sqrt {a^2 - x^2} }\right)^3} 3 - a^2 \sqrt {a^2 - x^2} + C$


Primitive of Reciprocal of x squared by Root of a squared minus x squared

:$\displaystyle \int \frac {\mathrm d x} {x^2 \sqrt {a^2 - x^2} } = \frac {-\sqrt {a^2 - x^2} } {a^2 x} + C$


Primitive of Reciprocal of x cubed by Root of a squared minus x squared

:$\displaystyle \int \frac {\d x} {x^3 \sqrt {a^2 - x^2} } = \frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$


Primitive of Root of a squared minus x squared

:$\displaystyle \int \sqrt {a^2 - x^2} \rd x = \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C$


Primitive of x by Root of a squared minus x squared

:$\displaystyle \int x \sqrt {a^2 - x^2} \ \mathrm d x = \frac {-\left({\sqrt {a^2 - x^2} }\right)^3} 3 + C$


Primitive of x cubed by Root of a squared minus x squared

:$\displaystyle \int x^3 \sqrt {a^2 - x^2} \ \mathrm d x = \frac {\left({\sqrt {a^2 - x^2} }\right)^5} 5 - \frac {a^2 \left({\sqrt {a^2 - x^2} }\right)^3} 3 + C$


Primitive of Root of a squared minus x squared over x

:$\displaystyle \int \frac {\sqrt {a^2 - x^2} } x \rd x = \sqrt {a^2 - x^2} - a \, \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$


Primitive of Root of a squared minus x squared over x squared

:$\displaystyle \int \frac {\sqrt {a^2 - x^2} } {x^2} \rd x = \frac {-\sqrt {a^2 - x^2} } x - \arcsin \frac x a + C$


Primitive of Reciprocal of Root of a squared minus x squared cubed

:$\displaystyle \int \frac {\mathrm d x} {\left({\sqrt {a^2 - x^2} }\right)^3} = \frac x {a^2 \sqrt {a^2 - x^2} } + C$


Primitive of x over Root of a squared minus x squared cubed

:$\displaystyle \int \frac {x \ \mathrm d x} {\left({\sqrt {a^2 - x^2} }\right)^3} = \frac 1 {\sqrt {a^2 - x^2} } + C$


Primitive of x squared over Root of a squared minus x squared cubed

:$\displaystyle \int \frac {x^2 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac x {\sqrt {a^2 - x^2} } - \arcsin \frac x a + C$


Primitive of Reciprocal of x by Root of a squared minus x squared cubed

:$\displaystyle \int \frac {\mathrm d x} {x \left({\sqrt {a^2 - x^2} }\right)^3} = \frac 1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right) + C$


Primitive of Reciprocal of x squared by Root of a squared minus x squared cubed

:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({\sqrt {a^2 - x^2} }\right)^3} = \frac {-\sqrt {a^2 - x^2} } {a^4 x} + \frac x {a^4 \sqrt {a^2 - x^2} } + C$


Primitive of Root of a squared minus x squared cubed

:$\displaystyle \int \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {3 a^2 x \sqrt {a^2 - x^2} } 8 + \frac {3 a^4} 8 \arcsin \frac x a + C$


Primitive of x by Root of a squared minus x squared cubed

:$\displaystyle \int x \left({\sqrt {a^2 - x^2} }\right)^3 \ \mathrm d x = \frac {-\left({\sqrt {a^2 - x^2} }\right)^5} 5 + C$


Primitive of x squared by Root of a squared minus x squared cubed

:$\displaystyle \int x^2 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {a^2 - x^2} }^3} {24} + \frac {a^4 x \sqrt {a^2 - x^2} } {16} + \frac {a^6} {16} \arcsin \frac x a + C$


Primitive of Reciprocal of a x squared plus b x plus c

:$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \begin{cases} \dfrac 2 {\sqrt {4 a c - b^2} } \, \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \dfrac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\dfrac {2 a x + b - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^2 - 4 a c > 0 \\ \dfrac {-2} {2 a x + b} + C & : b^2 = 4 a c \end{cases}$


Primitive of Reciprocal of a x squared plus b x plus c/a equal to 0

:$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \frac 1 b \ln \size {b x + c} + C$ when $a = 0$.


Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 2

Let $a \in \R_{\ne 0}$. Let $b = 0$. Then: :$\displaystyle \int \frac {\mathrm d x} {a x^2 + b x + c} = \begin{cases} \dfrac 1 {\sqrt {a c} } \arctan \left({x \sqrt {\dfrac a c} }\right) + C & : a c > 0 \\ \dfrac 1 {2 \sqrt {-a c} } \ln \left\vert{\dfrac {a x - \sqrt {-a c} } {a x + \sqrt {-a c} } }\right\vert + C & : a c < 0 \\ \dfrac {-1} {a x} + C & : c = 0 \end{cases}$


Primitive of x over a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x \ \mathrm d x} {a x^2 + b x + c} = \frac 1 {2 a} \ln \left\vert{a x^2 + b x + c}\right\vert - \frac b {2 a} \int \frac {\mathrm d x} {a x^2 + b x + c}$


Primitive of x squared over a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x^2 \rd x} {a x^2 + b x + c} = \frac x a - \frac b {2 a^2} \ln \size {a x^2 + b x + c} - \frac {b^2 - 2 a c} {2 a^2} \int \frac {\d x} {a x^2 + b x + c}$


Primitive of Reciprocal of a x squared plus b x plus c/Negative Discriminant

Let $a \in \R_{\ne 0}$. Let $b^2 - 4 a c < 0$. Then: :$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \frac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$


Primitive of Reciprocal of a x squared plus b x plus c/Positive Discriminant

Let $a \in \R_{\ne 0}$. Let $b^2 - 4 a c > 0$. Then: :$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \frac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\dfrac {2 a x + b - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C$


Primitive of Reciprocal of a x squared plus b x plus c/Zero Discriminant

Let $a \in \R_{\ne 0}$. Let $b^2 - 4 a c = 0$. Then: :$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \frac {-2} {2 a x + b} + C$


Primitive of Reciprocal of square of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^2} = \frac {2 a x + b} {\left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {2 a} {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}$


Completing the Square

Let $a, b, c, x$ be real numbers with $a \ne 0$. Then: :$a x^2 + b x + c = \dfrac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}$ This process is known as '''completing the square'''.


Primitive of x over square of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2} = \frac {- \left({b x + 2 c}\right)} {\left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} - \frac b {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}$


Primitive of x squared over square of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2} = \frac {\left({b^2 - 2 a c}\right) x + b c} {a \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {2 c} {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}$


Primitive of Power of x over Power of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: {{begin-eqn}} {{eqn | l = \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}       | r = \frac {x^{m - 1} } {\paren {2 n - m - 1} a \paren {a x^2 + b x + c}^{n - 1} }       | c =  }} {{eqn | o =       | ro= +       | r = \frac {\paren {m - 1} c} {\paren {2 n - m - 1} a} \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}       | c =  }} {{eqn | o =       | ro= -       | r = \frac {\paren {n - m} b} {\paren {2 n - m - 1} a} \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n}       | c =  }} {{end-eqn}}


Primitive of Reciprocal of x by square of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x \paren {a x^2 + b x + c}^2} = \frac 1 {2 c \paren {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c} }$


Primitive of Reciprocal of x squared by square of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2} = \frac {-1} {c x \paren {a x^2 + b x + c} } - \frac {3 a} c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac {2 b} c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}$


Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: {{begin-eqn}} {{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}       | r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} } }} {{eqn | o =       | ro= -       | r = \frac {\paren {m + 2 n - 3} a} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n} }} {{eqn | o =       | ro= -       | r = \frac {\paren {m - n + 2} b} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n} }} {{end-eqn}}


Ring of Sets Closed under Finite Union

Let $\mathcal R$ be a ring of sets. Let $A_1, A_2, \ldots, A_n \in \mathcal R$. Then: :$\displaystyle \bigcup_{j \mathop = 1}^n A_j \in \mathcal R$


Sigma-Ring contains Limit Superior of Sequence of Sets

Let $\mathcal R$ be a $\sigma$-ring. Let $\left \langle{A_n}\right \rangle_{n \mathop \in \N} \in \mathcal R$ be a sequence of sets in $\mathcal R$. Then: :$\displaystyle \limsup_{n \to \infty} \ A_n \in \mathcal R$


Gamma Function for Non-Negative Integer Argument

The Gamma function satisfies: :$\map \Gamma z = \dfrac {\map \Gamma {z + 1} } z$ for any $z$ which is not a nonpositive integer.


Gamma Function of Positive Half-Integer

{{begin-eqn}} {{eqn | l = \Gamma \left({m + \frac 1 2}\right)       | r = \frac {\left({2 m}\right)!} {2^{2 m} m!} \sqrt \pi       | c =  }} {{eqn | r = \frac {1 \times 3 \times 5 \times \cdots \times \left({2 m - 1}\right)} {2^m} \sqrt \pi       | c =  }} {{end-eqn}} where: : $m + \dfrac 1 2$ is a half-integer such that $m > 0$ : $\Gamma$ denotes the Gamma function.


Gamma Function of Negative Half-Integer

{{begin-eqn}} {{eqn | l = \map \Gamma {-m + \frac 1 2}       | r = \frac {\paren {-1}^m 2^{2 m} m!} {\paren {2 m}!} \sqrt \pi       | c =  }} {{eqn | r = \frac {\paren {-1}^m 2^m} {1 \times 3 \times 5 \times \cdots \times \paren {2 m - 1} } \sqrt \pi       | c =  }} {{end-eqn}} where: :$-m + \dfrac 1 2$ is a half-integer such that $m > 0$ :$\Gamma$ denotes the Gamma function.


Reciprocal times Derivative of Gamma Function

Let $\Gamma$ denote the Gamma function. Then: :$\displaystyle \dfrac {\Gamma\,' \left({z}\right)} {\Gamma \left({z}\right)} = -\gamma + \sum_{n \mathop = 1}^\infty \left({\frac 1 n - \frac 1 {z + n - 1} }\right)$ where: : $\Gamma\,' \left({z}\right)$ denotes the derivative of the Gamma function : $\gamma$ denotes the Euler-Mascheroni constant.


Derivative of Gamma Function at 1

Let $\Gamma$ denote the Gamma function. Then: :$\map {\Gamma'} 1 = -\gamma$ where: :$\map {\Gamma'} 1$ denotes the derivative of the Gamma function evaluated at $1$ :$\gamma$ denotes the Euler-Mascheroni constant.


Stirling's Formula for Gamma Function

Let $\Gamma$ denote the Gamma function. Then: :$\Gamma \left({x + 1}\right) = \sqrt {2 \pi x} \, x^x e^{-x} \left({1 + \dfrac 1 {12 x} + \dfrac 1 {288 x^2} - \dfrac {139} {51 \, 480 x^3} + \cdots}\right)$


Beta Function as Integral of Power of t over Power of t plus 1

:$\displaystyle \map \Beta {x, y} = \int_{\mathop \to 0}^{\mathop \to \infty} \frac {t^{x - 1} } {\paren {1 + t}^{x + y} } \rd t$ where $\Beta$ denotes the Beta function.


Beta Function as Integral of Power of t by Power of 1 minus t over Power of r plus t

:$\displaystyle \Beta \left({x, y}\right) := r^y \left({r + 1}\right)^x \int_{\mathop \to 0}^{\mathop \to 1} \frac {t^{x - 1} \left({1 - t}\right)^{y - 1} } {\left({r + t}\right)^{x + y} } \rd t$ where $\Beta$ denotes the Beta function.


Separation of Variables/General Result

Suppose a first order ordinary differential equation can be expressible in this form: :$\map {g_1} x \map {h_1} y + \map {g_2} x \map {h_2} y \dfrac {\d y} {\d x} = 0$ Then the equation is said to '''have separable variables''', or '''be separable'''. Its general solution is found by solving the integration: :$\displaystyle \int \frac {\map {g_1} x} {\map {g_2} x} \rd x + \int \frac {\map {h_2} y} {\map {h_1} y} \rd y = C$


Sum of Arithmetic-Geometric Sequence

Let $\sequence {a_k}$ be an arithmetic-geometric sequence defined as: :$a_k = \paren {a + k d} r^k$ for $k = 0, 1, 2, \ldots, n - 1$ Then its closed-form expression is: :$\displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {a \paren {1 - r^n} } {1 - r} + \frac {r d \paren {1 - n r^{n - 1} + \paren {n - 1} r^n} } {\paren {1 - r}^2}$


Reduction Formula for Integral of Power of Cosine

Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Then: :$\displaystyle \int \cos^n x \rd x = \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n \int \cos^{n - 2} x \rd x$ is a reduction formula for $\displaystyle \int \cos^n x \rd x$.


Primitives of Rational Functions involving Power of a x + b

This page gathers together the primitives of some rational functions involving a general power of $a x + b$.


Primitives of Functions involving Root of a x + b

This page gathers together the primitives of some rational functions involving $\sqrt {a x + b}$.


Primitives of Functions involving Power of Root of a x + b


Primitives of Functions involving a x + b and p x + q

This page gathers together the primitives of some rational functions involving $a x + b$ and $p x + q$.


Primitives of Functions involving Root of a x + b and p x + q

This page gathers together the primitives of some functions involving $\sqrt{a x + b}$ and $p x + q$.


Primitives of Functions involving Root of a x + b and Root of p x + q

This page gathers together the primitives of some functions involving $\sqrt{a x + b}$ and $\sqrt{p x + q}$.


Primitives involving x squared plus a squared

This page gathers together the primitives of some expressions involving $x^2 + a^2$.


Primitives involving x squared plus a squared squared

This page gathers together the primitives of some expressions involving $\left({x^2 + a^2}\right)^2$.


Primitives involving Power of x squared plus a squared

This page gathers together the primitives of some expressions involving $\left({x^2 + a^2}\right)^n$.


Primitives involving x squared minus a squared

This page gathers together the primitives of some expressions involving $x^2 - a^2$.


Primitives involving x squared minus a squared squared

This page gathers together the primitives of some expressions involving $\left({x^2 - a^2}\right)^2$.


Primitives involving Power of x squared minus a squared

This page gathers together the primitives of some expressions involving $\left({x^2 - a^2}\right)^n$.


Primitives involving a squared minus x squared

This page gathers together the primitives of some expressions involving $a^2 - x^2$.


Primitives involving a squared minus x squared squared

This page gathers together the primitives of some expressions involving $\left({a^2 - x^2}\right)^2$.


Primitives involving Power of a squared minus x squared


Primitives involving Root of x squared plus a squared

This page gathers together the primitives of some expressions involving $\sqrt{x^2 + a^2}$.


Primitives involving Root of x squared plus a squared cubed

This page gathers together the primitives of some expressions involving $\paren {\sqrt {x^2 + a^2} }^3$.


Primitives involving Root of x squared minus a squared

This page gathers together the primitives of some expressions involving $\sqrt{x^2 - a^2}$.


Primitives involving Root of x squared minus a squared cubed

This page gathers together the primitives of some expressions involving $\left({\sqrt{x^2 - a^2} }\right)^3$.


Primitives involving Root of a squared minus x squared

This page gathers together the primitives of some expressions involving $\sqrt {a^2 - x^2}$.


Primitives involving Root of a squared minus x squared cubed

This page gathers together the primitives of some expressions involving $\left({\sqrt{a^2 - x^2} }\right)^3$.


Primitives involving a x squared plus b x plus c

This page gathers together the primitives of some expressions involving $a x^2 + b x + c$.


Primitive of Reciprocal of Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \begin{cases} \dfrac 1 {\sqrt a} \map \ln {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C & : b^2 - 4 a c > 0 \\ \dfrac 1 {\sqrt a} \map {\sinh^{-1} } {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \dfrac 1 {\sqrt a} \ln \size {2 a x + b} + C & : b^2 - 4 a c = 0 \end{cases}$


Primitive of Reciprocal of Root of a x squared plus b x plus c/Negative Discriminant

Let $a \in \R_{\ne 0}$. Let $b^2 - 4 a c < 0$. Then: :$\displaystyle \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \sinh^{-1} \paren {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$


Primitive of Reciprocal of Root of a x squared plus b x plus c/Positive Discriminant

Let $a \in \R_{\ne 0}$. Let $b^2 - 4 a c > 0$. Then: :$\displaystyle \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \map \ln {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C$


Primitive of Reciprocal of Root of a x squared plus b x plus c/Zero Discriminant

Let $a \in \R_{\ne 0}$. Let $b^2 - 4 a c = 0$. Then: :$\displaystyle \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \ln \left\vert{2 a x + b}\right\vert + C$


Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 1

Let $a \in \R_{\ne 0}$. Let $b = 0$. Then: :$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \begin {cases} \dfrac 1 {\sqrt {a c} } \map \arctan {x \sqrt {\dfrac a c} } + C & : a c > 0 \\ \dfrac 1 {2 \sqrt {-a c} } \ln \size {\dfrac {a x - \sqrt {-a c} } {a x + \sqrt {-a c} } } + C & : a c < 0 \\ \dfrac {-1} {a x} + C & : c = 0 \end {cases}$


Primitive of Reciprocal of x by Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = \begin{cases} \dfrac {-1} {\sqrt c} \ln \left({\dfrac {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} x}\right) & : b^2 - 4 a c > 0 \\ \dfrac {-1} {\sqrt c} \sinh^{-1} \left({\dfrac {b x + 2 c} {\left\vert{x}\right\vert \sqrt {4 a c - b^2} } }\right) & : b^2 - 4 a c < 0 \\ \dfrac {-1} {\sqrt c} \ln \left\vert{\dfrac {2 c} x + b}\right\vert + C & : b^2 - 4 a c = 0 \end{cases}$


Primitive of Reciprocal of x squared by Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x^2 \sqrt {a x^2 + b x + c} } = -\frac {\sqrt {a x^2 + b x + c} } {c x} - \frac b {2 c} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$


Primitive of Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$


Primitive of x by Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int x \sqrt {a x^2 + b x + c} \ \mathrm d x = \frac {\left({\sqrt {a x^2 + b x + c} }\right)^3} {3 a} - \frac {b \left({2 a x + b}\right) \sqrt {a x^2 + b x + c} } {8 a^2} - \frac {b \left({4 a c - b^2}\right)} {16 a^2} \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} }$


Primitive of x squared by Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int x^2 \sqrt {a x^2 + b x + c} \rd x = \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x$


Primitive of Root of a x squared plus b x plus c over x

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\sqrt {a x^2 + b x + c} } x \rd x = \sqrt {a x^2 + b x + c} + \frac b 2 \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$


Primitive of Root of a x squared plus b x plus c over x squared

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\sqrt {a x^2 + b x + c} } {x^2} \ \mathrm d x = \frac {-\sqrt {a x^2 + b x + c} } x + a \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} } + \frac b 2 \int \frac {\mathrm d x} {x \sqrt {a x^2 + b x + c} }$


Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\mathrm d x} {\left({\sqrt {a x^2 + b x + c} }\right)^3} = \frac {2 \left({2 a x + b}\right)} {\left({4 a c - b^2}\right) \sqrt {a x^2 + b x + c} } + C$


Primitive of x over Cube of Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x \ \mathrm d x} {\left({\sqrt {a x^2 + b x + c} }\right)^3} = \frac {2 \left({b x + 2 c}\right)} {\left({b^2 - 4 a c}\right) \sqrt {a x^2 + b x + c} }$


Primitive of Reciprocal of x by Cube of Root of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\mathrm d x} {x \left({\sqrt {a x^2 + b x + c} }\right)^3} = \frac 1 {c \sqrt {a x^2 + b x + c} } + \frac 1 c \int \frac {\mathrm d x} {x \sqrt {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\mathrm d x} {\left({\sqrt {a x^2 + b x + c} }\right)^3}$


Primitive of x by Half Integer Power of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int x \left({a x^2 + b x + c}\right)^{n + \frac 1 2} \ \mathrm d x = \frac {\left({a x^2 + b x + c}\right)^{n + \frac 3 2} } {a \left({2 n + 3}\right)} - \frac b {2 a} \int \left({a x^2 + b x + c}\right)^{n + \frac 1 2} \ \mathrm d x$


Primitive of x over x cubed plus a cubed

:$\displaystyle \int \frac {x \rd x} {x^3 + a^3} = \frac 1 {6 a} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {a \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$


Primitive of Reciprocal of x cubed plus a cubed

:$\displaystyle \int \frac {\d x} {x^3 + a^3} = \frac 1 {6 a^2} \ln \size {\frac {\paren {x + a}^2} {x^2 - a x + a^2} } + \frac 1 {a^2 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$


Primitive of Reciprocal of x by x cubed plus a cubed

:$\displaystyle \int \frac {\mathrm d x} {x \left({x^3 + a^3}\right)} = \frac 1 {3 a^3} \ln \left\vert{\frac {x^3} {x^3 + a^3} }\right\vert + C$


Primitive of Reciprocal of x cubed plus a cubed squared

:$\displaystyle \int \frac {\mathrm d x} {\left({x^3 + a^3}\right)^2} = \frac x {3 a^3 \left({x^3 + a^3}\right)} + \frac 1 {9 a^5} \ln \left({\frac {\left({x + a}\right)^2} {x^2 - a x + a^2} }\right) + \frac 2 {3 a^5 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$


Primitive of x squared over x cubed plus a cubed squared

:$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({x^3 + a^3}\right)^2} = \frac {-1} {3 \left({x^3 + a^3}\right)} + C$