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

:$\displaystyle \int \frac {\mathrm d x} {x \left({x^3 + a^3}\right)^2} = \frac 1 {3 a^3 \left({x^3 + a^3}\right)} + \frac 1 {3 a^6} \ln \left\vert{\frac {x^3} {x^3 + a^3} }\right\vert$


Primitive of Power of x over x cubed plus a cubed

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


Primitive of Reciprocal of x fourth plus a fourth

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


Primitive of x over x fourth plus a fourth

:$\displaystyle \int \frac {x \rd x} {x^4 + a^4} = \frac 1 {2 a^2} \arctan \frac {x^2} {a^2}$


Primitive of x squared over x fourth plus a fourth

:$\displaystyle \int \frac {x^2 \rd x} {x^4 + a^4} = \frac 1 {4 a \sqrt 2} \map \ln {\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} } - \frac 1 {2 a \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$


Primitive of x cubed over x fourth plus a fourth

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


Primitive of Reciprocal of x by x fourth plus a fourth

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


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

:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({x^4 + a^4}\right)} = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^6} \arctan \frac {x^2} {a^2}$


Primitive of x over x fourth minus a fourth

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


Primitive of x squared over x fourth minus a fourth

:$\displaystyle \int \frac {x^2 \rd x} {x^4 - a^4} = \frac 1 {4 a} \ln \size {\frac {x - a} {x + a} } + \frac 1 {2 a} \arctan \frac x a + C$


Primitive of x cubed over x fourth minus a fourth

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


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

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


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

:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({x^4 - a^4}\right)} = \frac 1 {2 a^4 x^2} + \frac 1 {4 a^6} \ln \left\vert{\frac {x^2 - a^2} {x^2 + a^2} }\right\vert + C$


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

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


Absolute Value of Even Power

Let $x \in \R$ be a real number. Let $n \in \Z$ be an integer. Then: :$\left\vert{x^{2 n} }\right\vert = x^{2 n}$


Primitive of Power of x less one over Power of x plus Power of a

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


Primitive of x cubed over x fourth plus a fourth/Proof 1


Primitive of x cubed over x fourth plus a fourth/Proof 2


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

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


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

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


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

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


Primitive of Power of x less one over Power of x minus Power of a

:$\displaystyle \int \frac {x^{n - 1} \rd x} {x^n - a^n} = \frac 1 n \ln \size {x^n - a^n} + C$


Primitive of x by Sine of a x

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


Primitive of x squared by Sine of a x

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


Primitive of Sine of a x over x squared

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


Primitive of x over Sine of a x

{{begin-eqn}} {{eqn | l = \int \frac {x \rd x} {\sin a x}       | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C       | c =  }} {{eqn | r = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} \cdots} + C       | c =  }} {{end-eqn}}


Primitive of Square of Sine of a x

:$\displaystyle \int \sin^2 a x \rd x = \frac x 2 - \frac {\sin 2 a x} {4 a} + C$


Primitive of x by Square of Sine of a x

:$\ds \int x \sin^2 a x \rd x = \frac {x^2} 4 - \frac {x \sin 2 a x} {4 a} - \frac {\cos 2 a x} {8 a^2} + C$


Primitive of Cube of Sine of a x

:$\displaystyle \int \sin^3 a x \rd x = -\frac {\cos a x} a + \frac {\cos^3 a x} {3 a} + C$


Primitive of Square of Cosecant of a x

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


Primitive of Reciprocal of Square of Sine of a x

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


Primitive of Reciprocal of Cube of Sine of a x

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


Primitive of Sine of p x by Sine of q x

For $p \ne q$: :$\displaystyle \int \sin p x \sin q x \rd x = \frac {\map \sin {p - q} x} {2 \paren {p - q} } - \frac {\map \sin {p + q} x} {2 \paren {p + q} } + C$


Primitive of Reciprocal of 1 minus Sine of a x

:$\displaystyle \int \frac {\d x} {1 - \sin a x} = \frac 1 a \tan \left({\frac \pi 4 + \frac {a x} 2}\right) + C$


Primitive of x over 1 minus Sine of a x

:$\ds \int \frac {x \rd x} {1 - \sin a x} = \frac x a \map \tan {\frac \pi 4 + \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 - \frac {a x} 2} } + C$


Primitive of Reciprocal of 1 plus Sine of a x

:$\displaystyle \int \frac {\d x} {1 + \sin a x} = -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C$


Primitive of x over 1 plus Sine of a x

:$\ds \int \frac {x \rd x} {1 + \sin a x} = -\frac x a \map \tan {\frac \pi 4 - \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 + \frac {a x} 2} } + C$


Reciprocal of One Minus Sine

:$\dfrac 1 {1 - \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 + \dfrac x 2}$


Reciprocal of One Plus Sine

:$\dfrac 1 {1 + \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 - \dfrac x 2}$


Power Series Expansion for Cosecant Function

The cosecant function has a Laurent series expansion: {{begin-eqn}} {{eqn | l = \csc x       | r = \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \,  x^{2 n - 1} } {\paren {2 n}!}       | c =  }} {{eqn | r = \frac 1 x + \frac x 6 + \frac {7 x^3} {360} + \frac {31 x^5} {15 \, 120} + \cdots       | c =  }} {{end-eqn}} where $B_n$ denotes the Bernoulli numbers. This converges for $0 < \size x < \pi$.


Periodic Function plus Constant

Let $f: \mathbb F \to \mathbb F$ be a function, where $\mathbb F \in \left\{{\R, \C}\right\}$. Let $k \in \mathbb F$ be constant. Then $f$ is periodic with period $L$ {{iff}} $f + k$ is periodic with period $L$.


Sum of Two Odd Powers/Examples/Sum of Two Cubes

:$x^3 + y^3 = \paren {x + y} \paren {x^2 - x y + y^2}$


Primitive of Periodic Function

Let $f: \R \to \R$ be a real function. Let $F$ be a primitive of $f$ that is bounded on all of $\R$. Let $f$ be periodic with period $L$. Then $F$ is also periodic with period $L$.


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

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


Sum of Two Fourth Powers

:$x^4 + y^4 = \paren {x^2 + \sqrt 2 x y + y^2} \paren {x^2 - \sqrt 2 x y + y^2}$


One Plus Tangent Half Angle over One Minus Tangent Half Angle

:$\dfrac {1 + \tan \frac x 2} {1 - \tan \frac x 2} = \sec x + \tan x$


Primitive of Power of Cosecant of a x

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


Primitive of Cosecant of a x/Tangent Form

:$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$ where $\tan \dfrac {a x} 2 \ne 0$.


Primitive of Reciprocal of Square of 1 plus Sine of a x

:$\displaystyle \int \frac {\mathrm d x} {\left({1 + \sin a x}\right)^2} = \frac {-1} {2a} \tan \left({\frac \pi 4 - \frac {a x} 2}\right) - \frac 1 {6 a} \tan^3 \left({\frac \pi 4 - \frac {a x} 2}\right) + C$


Primitive of Reciprocal of p plus q by Sine of a x

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


Primitive of Reciprocal of square of p plus q by Sine of a x

:$\displaystyle \int \frac {\d x} {\paren {p + q \sin a x}^2} = \frac {q \cos a x} {a \paren {p^2 - q^2} \paren {p + q \sin a x} } + \frac p {p^2 - q^2} \int \frac {\d x} {p + q \sin a x}$


Primitive of Reciprocal of p squared plus square of q by Sine of a x

:$\displaystyle \int \frac {\d x} {p^2 + q^2 \sin^2 a x} = \frac 1 {a p \sqrt {p^2 + q^2} } \arctan \frac {\sqrt {p^2 + q^2} \tan a x} p + C$


Primitive of Sine of a x over Power of x

:$\displaystyle \int \frac {\sin a x} {x^n} \rd x = \frac {-\sin a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cos a x} {x^{n - 1} } \rd x$


Primitive of Reciprocal of Power of Sine of a x

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


Primitive of x over Power of Sine of a x

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


Primitive of x by Cosine of a x

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


Primitive of x squared by Cosine of a x

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


Primitive of x cubed by Cosine of a x

:$\displaystyle \int x^3 \map \cos {a x} \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \cos a x + \paren {\frac {x^3} a - \frac {6 x} {a^3} } \sin a x + C$


Primitive of Reciprocal of Cosine of a x/Logarithm of Tangent Form

:$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$


Primitive of Square of Cosine of a x

:$\displaystyle \int \cos^2 a x \, \mathrm d x = \frac x 2 + \frac {\sin 2 a x} {4 a} + C$


Primitive of x by Square of Cosine of a x

:$\displaystyle \int x \cos^2 a x \ \mathrm d x = \frac {x^2} 4 + \frac {x \sin 2 a x} {4 a} + \frac {\cos 2 a x} {8 a^2} + C$


Primitive of Fourth Power of Cosine of a x

:$\displaystyle \int \cos^4 a x \ \mathrm d x = \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$


Primitive of Reciprocal of Square of Cosine of a x

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


Primitive of Square of Secant of a x

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


Primitive of Secant of a x/Tangent Form

:$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$ where $\map \tan {\dfrac \pi 4 + \dfrac {a x} 2} \ne 0$.


Primitive of Secant of a x/Secant plus Tangent Form

:$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\sec a x + \tan a x} + C$ where $\sec a x + \tan a x \ne 0$.


Primitive of Power of Secant of a x

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


Primitive of Reciprocal of Cube of Cosine of a x

:$\displaystyle \int \frac {\d x} {\cos^3 a x} = \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$


Primitive of Cosine of a x by Cosine of p x

:$\displaystyle \int \cos a x \cos p x \rd x = \frac {\map \sin {\paren {a - p} x} } {2 \paren {a - p} } + \frac {\map \sin {\paren {a + p} x} } {2 \paren {a + p} } + C$


Primitive of Reciprocal of 1 minus Cosine of a x

:$\displaystyle \int \frac {\mathrm d x} {1 - \cos a x} = \frac {-1} a \cot \frac {a x} 2 + C$


Primitive of x over 1 minus Cosine of a x

:$\displaystyle \int \frac {x \rd x} {1 - \cos a x} = \frac {-x} a \cot \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sin \frac {a x} 2} + C$


Primitive of Reciprocal of 1 plus Cosine of a x

:$\displaystyle \int \frac {\mathrm d x} {1 + \cos a x} = \frac 1 a \tan \frac {a x} 2 + C$


Primitive of Reciprocal of p plus q by Cosine of a x

:$\displaystyle \int \frac {\mathrm d x} {p + q \cos a x} = \begin{cases}  \displaystyle \frac 2 {a \sqrt {p^2 - q^2} } \arctan \left({\sqrt {\frac {p - q} {p + q} } \tan \dfrac {a x} 2}\right) + C & : p^2 > q^2 \\ \displaystyle \frac 1 {a \sqrt {q^2 - p^2} } \ln \left\vert{\frac {\tan \dfrac {a x} 2 + \sqrt {\dfrac {q + p} {q - p} } } {\tan \dfrac {a x} 2 - \sqrt {\dfrac {q + p} {q - p} } } }\right\vert + C & : p^2 < q^2 \\ \end{cases}$


Primitive of Power of x by Cosine of a x

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


Primitive of Reciprocal of Power of Cosine of a x

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


Primitive of x over Power of Cosine of a x

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


Primitive of Sine of a x by Cosine of a x

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


Primitive of Sine of p x by Cosine of q x

:$\displaystyle \int \sin p x \cos q x \rd x = \frac {-\cos \left({p - q}\right) x} {2 \left({p - q}\right)} - \frac {\cos \left({p + q}\right) x} {2 \left({p + q}\right)} + C$ for $p, q \in \R: p \ne q$


Primitive of Power of Sine of a x by Cosine of a x

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


Primitive of Power of Cosine of a x by Sine of a x

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


Primitive of Sine of a x squared by Cosine of a x squared

:$\displaystyle \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$


Primitive of Reciprocal of Sine of a x by Cosine of a x

:$\displaystyle \int \frac {\mathrm d x} {\sin a x \cos a x} = \frac 1 a \ln \left\vert{\tan a x}\right\vert + C$


Primitive of Square of Secant of a x over Tangent of a x

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


Primitive of Reciprocal of Square of Sine of a x by Cosine of a x

:$\displaystyle \int \frac {\d x} {\sin^2 a x \cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } - \frac 1 {a \sin a x} + C$


Primitive of Power of Cosecant of a x by Cotangent of a x

:$\displaystyle \int \csc^n a x \cot a x \ \mathrm d x = \frac {-\csc^n a x} {n a} + C$


Primitive of Square of Sine of a x over Cosine of a x

:$\displaystyle \int \frac {\sin^2 a x \rd x} {\cos a x} = \frac {-\sin a x} a + \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$


Primitive of Square of Cosine of a x over Sine of a x

:$\displaystyle \int \frac {\cos^2 a x \ \mathrm d x} {\sin a x} = \frac {\cos a x} a + \frac 1 a \ln \left\vert{\tan \frac {a x} 2}\right\vert + C$


Primitive of Reciprocal of Cosine of a x by 1 plus Sine of a x

:$\displaystyle \int \frac {\mathrm d x} {\cos a x \left({1 + \sin a x}\right)} = \frac {-1} {2 a \left({1 + \sin a x}\right)} + \frac 1 {2 a} \ln \left\vert{\tan \left({\frac {a x} 2 + \frac \pi 4}\right)}\right\vert + C$


Primitive of Reciprocal of Cosine of a x by 1 minus Sine of a x

:$\displaystyle \int \frac {\d x} {\cos a x \paren {1 - \sin a x} } = \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$


Primitive of Reciprocal of Sine of a x by 1 plus Cosine of a x

:$\displaystyle \int \frac {\d x} {\sin a x \paren {1 + \cos a x} } = \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$


Primitive of Reciprocal of Sine of a x plus Cosine of a x

:$\displaystyle \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$


Primitive of Reciprocal of Sine of a x minus Cosine of a x

:$\displaystyle \int \frac {\mathrm d x} {\sin a x - \cos a x} = \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac {a x} 2 - \frac \pi 8}\right)}\right\vert + C$


Primitive of Sine of a x over Sine of a x plus Cosine of a x

:$\displaystyle \int \frac {\sin a x \ \mathrm d x} {\sin a x + \cos a x} = \frac x 2 - \frac 1 {2 a} \ln \left\vert{\sin a x + \cos a x}\right\vert + C$


Primitive of Sine of a x over Sine of a x minus Cosine of a x

:$\displaystyle \int \frac {\sin a x \rd x} {\sin a x - \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x - \cos a x} + C$


Primitive of Cosine of a x over Sine of a x plus Cosine of a x

:$\displaystyle \int \frac {\cos a x \rd x} {\sin a x + \cos a x} = \frac x 2 + \frac 1 {2 a} \ln \size {\sin a x + \cos a x} + C$


Primitive of Cosine of a x over Sine of a x minus Cosine of a x

:$\displaystyle \int \frac {\cos a x \ \mathrm d x} {\sin a x - \cos a x} = \frac {-x} 2 + \frac 1 {2 a} \ln \left\vert{\sin a x - \cos a x}\right\vert + C$


Primitive of Sine of a x over p plus q of Cosine of a x

:$\displaystyle \int \frac {\sin a x \rd x} {p + q \cos a x} = \frac {-1} {a q} \ln \size {p + q \cos a x} + C$


Primitive of Sine of a x over Power of p plus q of Cosine of a x

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


Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x

:$\displaystyle \int \frac {\mathrm d x} {p \sin a x + q \cos a x} = \frac 1 {a \sqrt {p^2 + q^2} } \ln \tan \left\vert{\frac {a x + \arctan \dfrac q p} 2}\right\vert + C$


Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x plus r

:$\ds \int \frac {\d x} {p \sin a x + q \cos a x + r} = \begin{cases} \ds \frac 2 {a \sqrt {r^2 - p^2 - q^2} } \map \arctan {\frac {p + \paren {r - q} \tan \dfrac {a x} 2} {\sqrt {r^2 - p^2 - q^2} } } + C & : p^2 + q^2 < r^2 \\ \ds \frac 1 {a \sqrt {p^2 + q^2 - r^2} } \ln \size {\frac {p - \sqrt {p^2 + q^2 - r^2} + \paren {r - q} \tan \dfrac {a x} 2} {p + \sqrt {p^2 + q^2 - r^2} + \paren {r - q} \tan \dfrac {a x} 2} } + C & : p^2 + q^2 > r^2 \end{cases}$


Primitive of Reciprocal of p by Sine of a x plus q by 1 plus Cosine of a x

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


Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x

:$\displaystyle \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x} = \frac 1 {a p q} \map \arctan {\frac {p \tan a x} q} + C$


Primitive of Reciprocal of p squared by square of Sine of a x minus q squared by square of Cosine of a x

:$\displaystyle \int \frac {\d x} {p^2 \sin^2 a x - q^2 \cos^2 a x} = \frac 1 {2 a p q} \ln \size {\frac {p \tan a x - q} {p \tan a x + q} } + C$


Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Sine

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


Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Cosine

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


Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Both Powers

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


Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Cosine

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


Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Both Powers

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


Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Sine

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


Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Cosine

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


Primitive of Tangent of a x/Cosine Form

:$\ds \int \tan a x \rd x = \frac {-\ln \size {\cos a x} } a + C$


Primitive of Square of Tangent of a x

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


Primitive of Cube of Tangent of a x

:$\displaystyle \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$


Primitive of Power of Tangent of a x by Square of Secant of a x

:$\displaystyle \int \tan^n a x \sec^2 a x \rd x = \frac {\tan^{n + 1} a x} {\paren {n + 1} a} + C$


Primitive of Cotangent of a x

:$\displaystyle \int \cot a x \rd x = \frac {\ln \size {\sin a x} } a + C$


Primitive of x by Tangent of a x

:$\displaystyle \int x \tan a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x} ^ 3} 3 + \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$ where $B_{2 n}$ denotes the $2 n$th Bernoulli number.


Primitive of Tangent of a x over x

:$\displaystyle \int \frac {\tan a x} x \ \mathrm d x = a x + \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$ where $B_n$ denotes the $n$th Bernoulli number.


Primitive of Reciprocal of p plus q by Tangent of a x

:$\displaystyle \int \frac {\d x} {p + q \tan a x} = \frac {p x} {p^2 + q^2} + \frac q {a \paren {p^2 + q^2} } \ln \size {q \sin a x + p \cos a x} + C$


Primitive of Square of Cotangent of a x

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


Primitive of Cube of Cotangent of a x

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


Primitive of Power of Cotangent of a x by Square of Cosecant of a x

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


Primitive of Square of Cosecant of a x over Cotangent of a x

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


Primitive of Cotangent of a x over x

:$\displaystyle \int \frac {\cot a x} x \ \mathrm d x = \frac {-1} a x - \frac {a x} 3 - \frac {\paren {a x}^3} {135} - \cdots - \frac {\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} - \cdots + C$ where $B_n$ denotes the $n$th Bernoulli number.


Primitive of x by Square of Cotangent of a x

:$\displaystyle \int x \cot^2 a x \ \mathrm d x = \frac {-x \cot a x} a + \frac 1 {a^2} \ln \left\vert{\sin a x}\right\vert - \frac {x^2} 2 + C$


Primitive of Power of Cotangent of a x

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


Primitive of Cube of Secant of a x

:$\displaystyle \int \sec^3 a x \rd x = \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C$


Primitive of x by Secant of a x

{{begin-eqn}} {{eqn | l = \int x \sec a x \rd x       | r = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots + \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + \cdots} + C       | c =  }} {{eqn | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C       | c =  }} {{end-eqn}} where $E_{2 n}$ is the $2 n$th Euler number.


Primitive of Secant of a x over x

:$\displaystyle \int \frac {\sec a x} x \ \mathrm d x = \ln \size x + \frac {\paren {a x}^2} 4 + \frac {5 \paren {a x}^4} {96} + \frac {61 \paren {a x}^6} {4320} + \cdots + \frac {\paren {-1}^n E_n \paren {a x}^{2 n} } {\paren {2 n} \paren {2 n}!} + \cdots + C$ where $E_n$ is the $n$th Euler number.


Primitive of x by Square of Secant of a x

:$\displaystyle \int x \sec^2 a x \ \mathrm d x = \frac {x \tan a x} a + \frac  1 {a^2} \ln \left\vert{\cos a x}\right\vert + C$


Primitive of Reciprocal of q plus p by Secant of a x

:$\ds \int \frac {\d x} {q + p \sec a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \cos a x} + C$


Primitive of x by Cosecant of a x

:$\displaystyle \int x \csc a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_n \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$ where $B_{2 n}$ is the $2 n$th Bernoulli number.


Primitive of Cosecant of a x over x

:$\displaystyle \int \frac {\csc a x} x \rd x = \frac {-1} {a x} + \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$ where $B_n$ is the $n$th Bernoulli number.


Primitive of x by Square of Cosecant of a x

:$\displaystyle \int x \csc^2 a x \ \mathrm d x = \frac {-x \cot a x} a + \frac  1 {a^2} \ln \left\vert{\sin a x}\right\vert + C$


Primitive of Reciprocal of q plus p by Cosecant of a x

:$\displaystyle \int \frac {\mathrm d x} {q + p \csc a x} = \frac x q - \frac p q \int \frac {\mathrm d x} {p + q \sin a x} + C$


Primitive of Arcsine of x over a

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


Primitive of x squared by Arcsine of x over a

:$\displaystyle \int x^2 \arcsin \frac x a \rd x = \frac {x^3} 3 \arcsin \frac x a + \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$


Primitive of Arccosine of x over a

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


Primitive of x by Arccosine of x over a/Proof 1

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


Primitive of x by Arccosine of x over a/Proof 2

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


Primitive of x squared by Arccosine of x over a

:$\displaystyle \int x^2 \arccos \frac x a \rd x = \frac {x^3} 3 \arccos \frac x a - \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$


Primitive of Arccosine of x over a over x squared

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


Primitive of Square of Arccosine of x over a

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


Primitive of Arctangent of x over a

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


Primitive of x by Arctangent of x over a

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


Primitive of x by Arccotangent of x over a

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


Primitive of x squared by Arctangent of x over a

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


Primitive of Arctangent of x over a over x

{{begin-eqn}} {{eqn 	| l = \int \frac 1 x \arctan \paren {\frac x a} \rd x       	| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {x^{2 k + 1} } {\paren {2 k + 1}^2 a^{2 k + 1} }  }} {{eqn 	| r = \frac x a - \frac {x^3} {3^2 a^3} + \frac {x^5} {5^2 a^5} - \frac {x^7} {7^2 a^7} + \cdots + C }} {{end-eqn}}


Primitive of Arccotangent of x over a over x squared

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


Primitive of Arccotangent of x over a

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


Primitive of Arccotangent of x over a/Proof 1

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


Primitive of Arccotangent of x over a/Proof 2


Primitive of x by Arcsecant of x over a

:$\displaystyle \int x \operatorname{arcsec} \frac x a \ \mathrm d x = \begin{cases} \displaystyle \frac {x^2} 2 \operatorname{arcsec} \frac x a - \frac {a \sqrt{x^2 - a^2} } 2 + C & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {x^2} 2 \operatorname{arcsec} \frac x a + \frac {a \sqrt{x^2 - a^2} } 2 + C & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}$


Primitive of x by Arccosecant of x over a

:$\displaystyle \int x \arccsc \frac x a \rd x = \begin{cases} \displaystyle \frac {x^2} 2 \arccsc \frac x a + \frac {a \sqrt {x^2 - a^2} } 2 + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {x^2} 2 \arccsc \frac x a - \frac {a \sqrt {x^2 - a^2} } 2 + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$


Primitive of x squared by Arcsecant of x over a

:$\displaystyle \int x^2 \operatorname{arcsec} \frac x a \ \mathrm d x = \begin{cases} \displaystyle \frac {x^3} 3 \operatorname{arcsec} \frac x a - \frac {a x \sqrt{x^2 - a^2} } 6 - \frac {a^3} 6 \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {x^3} 3 \operatorname{arcsec} \frac x a + \frac {a x \sqrt{x^2 - a^2} } 6 + \frac {a^3} 6 \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}$


Primitive of Arccosecant of x over a over x squared

:$\displaystyle \int \frac {\arccsc \frac x a} {x^2} \rd x = \begin{cases} \displaystyle \frac {-\arccsc \frac x a} x - \frac {\sqrt{x^2 - a^2} } {a x} + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {-\arccsc \frac x a} x + \frac {\sqrt{x^2 - a^2} } {a x} + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$


Primitive of Power of x by Arccosecant of x over a

:$\displaystyle \int x^m \arccsc \frac x a \rd x = \begin{cases} \displaystyle \frac {x^{m + 1} } {m + 1} \arccsc \frac x a + \frac a {m + 1} \int \frac {x^m \rd x} {\sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {x^{m + 1} } {m + 1} \arccsc \frac x a - \frac a {m + 1} \int \frac {x^m \rd x} {\sqrt {x^2 - a^2} } + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$


Primitive of Exponential of a x

:$\displaystyle \int e^{a x} \rd x = \frac {e^{a x} } a + C$


Primitive of x by Exponential of a x

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


Primitive of x squared by Exponential of a x

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


Primitive of Power of x by Exponential of a x

Let $n$ be a positive integer. Let $a$ be a non-zero real number. Then: {{begin-eqn}} {{eqn | l = \int x^n e^{a x} \rd x       | r = \frac {e^{a x} } a \paren {x^n - \dfrac {n x^{n - 1} } a + \dfrac {n \paren {n - 1} x^{n - 2} } {a^2} - \dfrac {n \paren {n - 1} \paren {n - 2} x^{n - 3} } {a^3} + \cdots + \dfrac {\paren {-1}^n n!} {a^n} } + C       | c =  }} {{eqn | r = \frac {e^{a x} } a \sum_{k \mathop = 0}^n \paren {\paren {-1}^k \frac {n^{\underline k} x^{n - k} } {a^k} } + C       | c =  }} {{end-eqn}} where $n^{\underline k}$ denotes the $k$th falling factorial power of $n$.


Primitive of Power of x by Exponential of a x/Lemma

Let $n$ be a positive integer. Then: :$\displaystyle \int x^n e^{a x} \rd x = \frac {x^n e^{a x} } a - \frac n a \int x^{n - 1} e^{a x} \rd x + C$


Primitive of Exponential of a x over x

:$\displaystyle \int \frac {e^{a x} \rd x} x = \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^k} {k \times k!} + C$


Primitive of Exponential of a x over Power of x

:$\displaystyle \int \frac {e^{a x} \rd x} {x^n} = \frac {-e^{a x} } {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {e^{a x} \rd x} {x^{n - 1} } + C$ where $n \ne 1$.


Primitive of Reciprocal of p plus q by Exponential of a x

:$\displaystyle \int \frac {\d x} {p + q e^{a x} } = \frac x p - \frac 1 {a p} \ln \size {p + q e^{a x} } + C$


Primitive of Reciprocal of p by Exponential of a x plus q by Exponential of -a x

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


Primitive of Exponential of a x by Sine of b x

:$\displaystyle \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$


Primitive of Exponential of a x by Cosine of b x

:$\displaystyle \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x}} {a^2 + b^2} + C$


Primitive of x by Exponential of a x by Sine of b x

:$\displaystyle \int x e^{a x} \sin b x \ \mathrm d x = \frac {x e^{a x} \left({a \sin b x - b \cos bx}\right)} {a^2 + b^2} - \frac {e^{a x} \left({\left({a^2 - b^2}\right) \sin b x - 2 a b \cos bx}\right)} {\left({a^2 + b^2}\right)^2} + C$


Primitive of Exponential of a x by Logarithm of x

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


Primitive of Exponential of a x by Power of Sine of b x

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


Primitive of Exponential of a x by Power of Cosine of b x

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


Primitive of Logarithm of x

:$\ds \int \ln x \rd x = x \ln x - x + C$


Primitive of x by Logarithm of x

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


Primitive of Power of x by Logarithm of x

:$\displaystyle \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \left({\ln x - \frac 1 {m + 1} }\right) + C$ where $m \ne -1$.


Primitive of Logarithm of x over x

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


Primitive of Square of Logarithm of x

:$\displaystyle \int \ln^2 x \ \mathrm d x = x \ln^2 x - 2 x \ln x + 2 x + C$


Primitive of Power of Logarithm of x over x

:$\displaystyle \int \frac {\ln^n x} x \ \mathrm d x = \frac {\ln^{n + 1} x} {n + 1} + C$


Primitive of Reciprocal of x by Logarithm of x

:$\displaystyle \int \frac {\mathrm d x} {x \ln x} = \ln \left({\ln x}\right) + C$


Primitive of Power of x over Logarithm of x

:$\displaystyle \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$


Primitive of Logarithm of x squared plus a squared

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


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

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


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

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


Primitive of Hyperbolic Sine of a x

:$\displaystyle \int \sinh a x \ \mathrm d x = \frac {\cosh a x} a + C$


Primitive of Hyperbolic Cosine of a x

:$\displaystyle \int \cosh a x \ \mathrm d x = \frac {\sinh a x} a + C$


Primitive of Hyperbolic Tangent of a x

:$\displaystyle \int \tanh a x \rd x = \frac {\ln \size {\cosh a x} } a + C$


Primitive of Hyperbolic Cotangent of a x

:$\displaystyle \int \coth a x \ \mathrm d x = \frac {\ln \left\vert{\sinh a x}\right\vert} a + C$


Primitive of Hyperbolic Cosecant of a x

:$\displaystyle \int \csch a x \rd x = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$


Primitive of x by Hyperbolic Sine of a x

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


Primitive of x by Hyperbolic Cosine of a x

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


Primitive of x squared by Hyperbolic Cosine of a x

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


Primitive of Hyperbolic Cosine of a x over x

{{begin-eqn}} {{eqn | l = \int \frac {\cosh a x \ \mathrm d x} x       | r = \ln \left\vert{x}\right\vert + \sum_{k \mathop \ge 1} \frac { \left({a x}\right)^{2 k} } {\left({2 k}\right) \left({2 k}\right)!} + C }} {{eqn | r = \ln \left\vert{x}\right\vert + \frac {\left({a x}\right)^2} {2 \times 2!} + \frac {\left({a x}\right)^4} {4 \times 4!} + \frac {\left({a x}\right)^6} {6 \times 6!} + \cdots + C }} {{end-eqn}}


Primitive of Hyperbolic Tangent of a x over x

{{begin-eqn}} {{eqn | l = \int \frac {\tanh a x \ \mathrm d x} x       | r = \sum_{k \mathop \ge 1} \frac {\left({-1}\right)^{k - 1} 2^{2 k} \left({2^{2 k} - 1}\right) B_k \left({a x}\right)^{2 k - 1} } {\left({2 k - 1}\right) \left({2 k}\right)!} + C }} {{eqn | r = a x - \frac {\left({a x}\right)^3} 9 + \frac {2 \left({a x}\right)^5} {75} - \cdots + C }} {{end-eqn}} where $B_k$ denotes the $k$th Bernoulli number.


Primitive of Hyperbolic Cotangent of a x over x

{{begin-eqn}} {{eqn | l = \int \frac {\coth a x \ \mathrm d x} x       | r = \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^k 2^{2 k} B_k \left({a x}\right)^{2 k - 1} } {\left({2 k - 1}\right) \left({2 k}\right)!} + C }} {{eqn | r = -\frac 1 {a x} + \frac {a x} 3 - \frac {\left({a x}\right)^3} {135} + \cdots + C }} {{end-eqn}} where $B_k$ denotes the $k$th Bernoulli number.


Primitive of Hyperbolic Secant of a x over x

{{begin-eqn}} {{eqn | l = \int \frac {\operatorname{sech} a x \ \mathrm d x} x       | r = \ln \left\vert{x}\right\vert + \sum_{k \mathop \ge 1} \frac {\left({-1}\right)^k E_k \left({a x}\right)^{2 k} } {\left({2 k}\right) \left({2 k}\right)!} + C }} {{eqn | r = \ln \left\vert{x}\right\vert - \frac {\left({a x}\right)^2} 4 + \frac {\left({a x}\right)^4} {96} - \frac {\left({a x}\right)^6} {4320} + \cdots + C }} {{end-eqn}} where $E_k$ denotes the $k$th Euler number.


Primitive of Hyperbolic Cosecant of a x over x

{{begin-eqn}} {{eqn | l = \int \frac {\operatorname{csch} a x \ \mathrm d x} x       | r = \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^k 2 \left({2^{2 k - 1} - 1}\right) B_k \left({a x}\right)^{2 k - 1} } {\left({2 k - 1}\right) \left({2 k}\right)!} + C }} {{eqn | r = - \frac 1 {a x} - \frac {a x} 6 + \frac {7 \left({a x}\right)^3} {1080} + \cdots + C }} {{end-eqn}} where $E_k$ denotes the $k$th Bernoulli number.


Derivative of Hyperbolic Sine of a x

:$\map {D_x} {\sinh a x} = a \cosh a x$


Derivative of Hyperbolic Cosine of a x

:$\map {\dfrac \d {\d x} } {\cosh a x} = a \sinh a x$


Primitive of Hyperbolic Sine of a x over x squared

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


Primitive of Reciprocal of Hyperbolic Sine of a x

:$\displaystyle \int \frac {\mathrm d x} {\sinh a x} = \frac 1 a \ln \left\vert {\tanh \frac {a x} 2} \right\vert + C$


Primitive of Reciprocal of Hyperbolic Tangent of a x

:$\displaystyle \int \frac {\mathrm d x} {\tanh a x} = \frac {\ln \left\vert {\sinh a x} \right\vert} a + C$


Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent

:$\coth x = \dfrac 1 {\tanh x}$ where $\tanh$ and $\coth$ denote hyperbolic tangent and hyperbolic cotangent respectively.


Primitive of Reciprocal of Hyperbolic Cotangent of a x

:$\displaystyle \int \frac {\mathrm d x} {\coth a x} = \frac {\ln \left\vert {\cosh a x} \right\vert} a + C$


Primitive of Reciprocal of Hyperbolic Cosecant of a x

:$\displaystyle \int \frac {\mathrm d x} {\operatorname{csch} a x} = \frac {\cosh a x} a + C$


Primitive of Square of Hyperbolic Sine of a x

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


Primitive of Square of Hyperbolic Cosine of a x

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


Primitive of Square of Hyperbolic Tangent of a x

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


Primitive of Square of Hyperbolic Cotangent of a x

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


Primitive of Square of Hyperbolic Secant of a x

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


Primitive of Square of Hyperbolic Cosecant of a x

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


Primitive of Square of Hyperbolic Sine of a x/Corollary

:$\displaystyle \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$


Primitive of x by Square of Hyperbolic Sine of a x

:$\ds \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$


Primitive of Square of Hyperbolic Cosine of a x/Corollary

:$\displaystyle \int \cosh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} + \frac x 2 + C$


Primitive of x by Square of Hyperbolic Cosine of a x

:$\displaystyle \int x \cosh^2 a x \ \mathrm d x = \frac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} + \frac {x^2} 4 + C$


Primitive of x by Square of Hyperbolic Tangent of a x

:$\displaystyle \int x \tanh^2 a x \ \mathrm d x = \frac {x^2} 2 - \frac {x \tanh a x} a + \frac 1 {a^2} \ln \left\vert{\cosh a x}\right\vert + C$


Primitive of x by Square of Hyperbolic Cotangent of a x

:$\displaystyle \int x \coth^2 a x \ \mathrm d x = \frac {x^2} 2 - \frac {x \coth a x} a + \frac 1 {a^2} \ln \left\vert{\sinh a x}\right\vert + C$


Primitive of x by Square of Hyperbolic Secant of a x

:$\displaystyle \int x \operatorname{sech}^2 a x \ \mathrm d x = \frac {x \tanh a x} a - \frac 1 {a^2} \ln \left\vert{\cosh a x}\right\vert + C$


Primitive of Hyperbolic Sine of a x by Hyperbolic Sine of p x

:$\displaystyle \int \sinh a x \sinh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$


Primitive of Hyperbolic Cosine of a x by Hyperbolic Cosine of p x

:$\displaystyle \int \cosh a x \cosh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } + \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$


Primitive of Hyperbolic Sine of a x by Sine of p x

:$\displaystyle \int \sinh a x \sin p x \ \mathrm d x = \frac {a \cosh a x \sin p x - p \sinh a x \cos p x} {a^2 + p^2} + C$


Primitive of Hyperbolic Cosine of a x by Sine of p x

:$\displaystyle \int \cosh a x \sin p x \ \mathrm d x = \frac {a \sinh a x \sin p x - p \cosh a x \cos p x} {a^2 + p^2} + C$


Primitive of Hyperbolic Cosine of a x by Cosine of p x

:$\displaystyle \int \cosh a x \cos p x \ \mathrm d x = \frac {a \sinh a x \cos p x + p \cosh a x \sin p x} {a^2 + p^2} + C$


Primitive of Reciprocal of p plus q by Hyperbolic Sine of a x

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


Primitive of Reciprocal of q plus p by Hyperbolic Secant of a x

:$\displaystyle \int \frac {\mathrm d x} {q + p \operatorname{sech} a x} = \frac x q - \frac p q \int \frac {\mathrm d x} {p + q \cosh a x} + C$


Primitive of Reciprocal of q plus p by Hyperbolic Cosecant of a x

:$\displaystyle \int \frac {\d x} {q + p \csch a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \sinh a x} + C$


Primitive of Power of x by Hyperbolic Cosine of a x

:$\ds \int x^m \cosh a x \rd x = \frac {x^m \sinh a x} a - \frac m a \int x^{m - 1} \sinh a x \rd x + C$


Primitive of Power of Hyperbolic Sine of a x

:$\displaystyle \int \sinh^n a x \rd x = \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x + C$


Primitive of Power of Hyperbolic Cosine of a x

:$\displaystyle \int \cosh^n a x \rd x = \frac {\cosh^{n - 1} a x \sinh a x} {a n} + \frac {n - 1} n \int \cosh^{n - 2} a x \rd x + C$


Primitive of Power of Hyperbolic Tangent of a x by Square of Hyperbolic Secant of a x

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


Derivative of Hyperbolic Tangent of a x

:$\map {\dfrac \d {\d x} } {\tanh a x} = a \sech^2 a x$


Derivative of Hyperbolic Cotangent of a x

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


Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x

:$\ds \int \coth^n a x \csch^2 a x \rd x = \frac {-\coth^{n + 1} a x} {\paren {n + 1} a} + C$


Primitive of Power of Hyperbolic Cotangent of a x

:$\ds \int \coth^n a x \rd x = \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C$


Derivative of Hyperbolic Secant of a x

:$\map {\dfrac \d {\d x} } {\sech a x} = -a \sech a x \tanh a x$


Derivative of Hyperbolic Cosecant of a x

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


Primitive of Power of Hyperbolic Secant of a x

:$\displaystyle \int \sech^n a x \rd x = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C$


Primitive of Power of Hyperbolic Cosecant of a x

:$\displaystyle \int \csch^n a x \rd x = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int\csch^{n - 2} a x \rd x + C$


Primitive of x over Power of Hyperbolic Sine of a x

:$\displaystyle \int \frac {x \rd x} {\sinh^n a x} = \frac {-x \cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sinh^{n - 2} a x} + C$


Primitive of x over Power of Hyperbolic Cosine of a x

:$\displaystyle \int \frac {x \rd x} {\cosh^n a x} = \frac {x \sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\cosh^{n - 2} a x} + C$


Primitive of Reciprocal of Hyperbolic Cosine of a x plus 1

:$\ds \int \frac {\d x} {\cosh a x + 1} = \frac 1 a \tanh \frac {a x} 2 + C$


Primitive of Reciprocal of Square of Hyperbolic Cosine of a x minus 1

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


Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

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


Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

:$\displaystyle \int \sinh^n a x \cosh a x \rd x = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C$ for $n \ne -1$.


Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x

:$\displaystyle \int \cosh^n a x \sinh a x \rd x = \frac {\cosh^{n + 1} a x} {\paren {n + 1} a} + C$ for $n \ne -1$.


Primitive of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x

:$\displaystyle \int \sinh^2 a x \cosh^2 a x \rd x = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C$


Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

:$\displaystyle \int \frac {\mathrm d x} {\sinh a x \cosh a x} = \frac 1 a \ln \left\vert{\tanh a x}\right\vert + C$


Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x

:$\displaystyle \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\tanh a x} = \frac 1 a \ln \left\vert{\tanh a x}\right\vert + C$


Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

:$\ds \int \frac {\d x} {\sinh^2 a x \cosh a x} = -\frac 1 a \map \arctan {\sinh a x} - \frac {\csch a x} a + C$


Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x

:$\displaystyle \int \sech^n a x \tanh a x \rd x = \frac {-\sech^n a x} {n a} + C$


Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x

:$\displaystyle \int \csch^n a x \coth a x \rd x = \frac {-\csch^n a x} {n a} + C$


Primitive of Square of Hyperbolic Sine of a x over Hyperbolic Cosine of a x

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


Primitive of Reciprocal of Hyperbolic Cosine of a x by 1 plus Hyperbolic Sine of a x

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


Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x minus 1

:$\displaystyle \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} } = \frac {-1} {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C$


Primitive of Cube of Hyperbolic Tangent of a x

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


Primitive of Cube of Hyperbolic Tangent of a x/Proof 1


Primitive of Cube of Hyperbolic Tangent of a x/Proof 2


Primitive of Cube of Hyperbolic Cotangent of a x/Proof 1

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


Primitive of Cube of Hyperbolic Cotangent of a x/Proof 2


Primitive of x by Hyperbolic Tangent of a x

:$\displaystyle \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$


Primitive of x by Hyperbolic Cotangent of a x

:$\displaystyle \int x \coth a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} + \cdots + \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$


Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x

:$\displaystyle \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$


Primitive of Cube of Hyperbolic Secant of a x

:$\displaystyle \int \sech^3 a x \rd x = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C$


Primitive of Cube of Hyperbolic Cosecant of a x

:$\ds \int \csch^3 a x \rd x = \frac {-\csch a x \coth a x} {2 a} - \frac 1 {2 a} \ln \size {\tanh a x} + C$


Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form

:$\ds \int \sech x \rd x =  \map \arctan {\sinh x} + C$


Primitive of Hyperbolic Secant of a x/Arcsine Form

:$\displaystyle \int \operatorname{sech} a x \ \mathrm d x = \frac {\arcsin \left({\tanh a x}\right)} a + C$


Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form

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


Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form

:$\displaystyle \int \operatorname{sech} a x \ \mathrm d x = \frac {\arctan \left({\sinh a x}\right)} a + C$


Primitive of x by Hyperbolic Cosecant of a x

:$\displaystyle \int x \csch a x \rd x = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C$


Derivative of Inverse Hyperbolic Sine of x over a

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


Primitive of Inverse Hyperbolic Sine of x over a

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


Derivative of Inverse Hyperbolic Cosine of x over a

:$\dfrac {\map \d {\map {\cosh^{-1} } {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 - a^2} }$ where $x > a$.


Derivative of Inverse Hyperbolic Tangent of x over a

:$\map {\dfrac \d {\d x} } {\map {\tanh^{-1} } {\dfrac x a} } = \dfrac a {a^2 - x^2}$ where $-a < x < a$.


Derivative of Inverse Hyperbolic Cotangent of x over a

:$\map {\dfrac \d {\d x} } {\map {\coth^{-1} } {\dfrac x a} } = \dfrac {-a} {x^2 - a^2}$ where $x^2 > a^2$.


Derivative of Inverse Hyperbolic Secant of x over a

:$\dfrac {\mathrm d \left({\operatorname{sech}^{-1} \left({\frac x a}\right)}\right)} {\mathrm d x} = \dfrac {-a} {x \sqrt{a^2 - x^2} }$ where $0 < x < a$.


Derivative of Inverse Hyperbolic Cosecant of x over a

:$\dfrac {\map \d {\csch^{-1} } {\frac x a} } {\d x} = \dfrac {-a} {\size x \sqrt {a^2 + x^2} }$ where $x \ne 0$.


Primitive of Inverse Hyperbolic Tangent of x over a

:$\displaystyle \int \tanh^{-1} \frac x a \rd x = x \tanh^{-1} \dfrac x a + \frac {a \ln \left({a^2 - x^2}\right)} 2 + C$


Primitive of Inverse Hyperbolic Cotangent of x over a

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


Primitive of Inverse Hyperbolic Secant of x over a

:$\displaystyle \int \sech^{-1} \frac x a \rd x = \begin{cases} x \sech^{-1} \dfrac x a + a \arcsin \dfrac x a + C & : \sech^{-1} \dfrac x a > 0 \\ x \sech^{-1} \dfrac x a - a \arcsin \dfrac x a + C & : \sech^{-1} \dfrac x a < 0 \end{cases}$


Inverse Hyperbolic Cosine of x over a in Logarithm Form

:$\cosh^{-1} \dfrac x a = \map \ln {x + \sqrt {x^2 - a^2} } - \ln a$


Inverse Hyperbolic Tangent of x over a in Logarithm Form

:$\tanh^{-1} \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} }$


Inverse Hyperbolic Cotangent of x over a in Logarithm Form

:$\coth^{-1} \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} }$


Inverse Hyperbolic Secant of x over a in Logarithm Form

:$\operatorname{sech}^{-1} \dfrac x a = \ln \left({\dfrac {a + \sqrt{a^2 - x^2} } x}\right)$


Inverse Hyperbolic Cosecant of x over a in Logarithm Form

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


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

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


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

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


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

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


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

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


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

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


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

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


Primitive of x squared over a squared minus x squared/Logarithm Form

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


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

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


Primitive of x by Inverse Hyperbolic Tangent of x over a

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


Primitive of x squared by Inverse Hyperbolic Sine of x over a

:$\displaystyle \int x^2 \sinh^{-1} \frac x a \rd x = \frac {x^3} 3 \sinh^{-1} \frac x a + \frac {\paren {2 a^2 - x^2} \sqrt {x^2 + a^2} } 9 + C$


Primitive of x squared by Inverse Hyperbolic Tangent of x over a

:$\displaystyle \int x^2 \tanh^{-1} \frac x a \ \mathrm d x = \frac {a x^2} 6 + \frac {x^3} 3 \tanh^{-1} \frac x a + \frac {a^3} 6 \ln \left({a^2 - x^2}\right) + C$


Primitive of Inverse Hyperbolic Sine of x over a over x squared

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


Primitive of Inverse Hyperbolic Cosine of x over a over x squared

:$\displaystyle \int \frac {\cosh^{-1} \dfrac x a \ \mathrm d x} {x^2} = \begin{cases} \displaystyle \frac {-\cosh^{-1} \dfrac x a} x - \frac 1 a \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) & : \cosh^{-1} \dfrac x a > 0 \\ \displaystyle \frac {-\cosh^{-1} \dfrac x a} x + \frac 1 a \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) & : \cosh^{-1} \dfrac x a < 0 \\ \end{cases}$


Primitive of Inverse Hyperbolic Tangent of x over a over x squared

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


Primitive of Inverse Hyperbolic Cotangent of x over a over x squared

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


Primitive of Power of x by Inverse Hyperbolic Sine of x over a

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


Primitive of Power of x by Inverse Hyperbolic Cotangent of x over a

:$\displaystyle \int x^m \coth^{-1} \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \coth^{-1} \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} } {a^2 - x^2} \rd x + C$


Primitive of Power of x by Inverse Hyperbolic Cosecant of x over a

:$\displaystyle \int x^m \operatorname{csch}^{-1} \frac x a \rd x = \begin{cases} \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a + \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x > 0 \\ \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a - \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x < 0 \\ \end{cases}$


Hyperbolic Tangent Half-Angle Substitution

:$\displaystyle \int \map F {\sinh x, \cosh x} \rd x = 2 \int \map F {\frac {2 u} {1 - u^2}, \frac {1 + u^2} {1 - u^2} } \frac {\d u} {1 - u^2}$ where $u = \tanh \dfrac x 2$.


Reciprocal of One Plus Cosine

:$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$


Reciprocal of One Minus Cosine

:$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \csc^2 \left({\dfrac x 2}\right)$


Irrationality of Logarithm

Let $a, b \in \N_{>0}$ such that both $\nexists m, n \in \N_{>0}: a^m = b^n$. Then $\log_b a$ is irrational.


Tangent of 22.5 Degrees

:$\tan 22.5^\circ = \tan \dfrac \pi 8 = \sqrt 2 - 1$


Tangent of 67.5 Degrees

:$\tan 67.5 \degrees = \tan \dfrac {3 \pi} 8 = \sqrt 2 + 1$


Sine of x plus Cosine of x/Cosine Form

:$\sin x + \cos x = \sqrt 2 \, \map \cos {x - \dfrac \pi 4}$


Sine of x minus Cosine of x/Sine Form

:$\sin x - \cos x = \sqrt 2 \sin \left({x - \dfrac \pi 4}\right)$


Sine of x minus Cosine of x

=== Sine Form === {{:Sine of x minus Cosine of x/Sine Form}} === Cosine Form === {{:Sine of x minus Cosine of x/Cosine Form}}


Sine of x minus Cosine of x/Cosine Form

:$\sin x - \cos x = \sqrt 2 \, \map \cos {x - \dfrac {3 \pi} 4}$


=== Cosine Form === {{:Multiple of Sine plus Multiple of Cosine/Cosine Form}} === Sine Form === {{:Multiple of Sine plus Multiple of Cosine/Sine Form}}


Sum Formulas for Sine and Cosine

=== Sine of x plus Cosine of x: Sine Form === {{:Sine of x plus Cosine of x/Sine Form}} === Sine of x plus Cosine of x: Cosine Form === {{:Sine of x plus Cosine of x/Cosine Form}} === Sine of x minus Cosine of x: Sine Form === {{:Sine of x minus Cosine of x/Sine Form}} === Sine of x minus Cosine of x: Cosine Form === {{:Sine of x minus Cosine of x/Cosine Form}} === Cosine of x minus Sine of x: Sine Form === {{:Cosine of x minus Sine of x/Sine Form}} === Cosine of x minus Sine of x: Cosine Form === {{:Cosine of x minus Sine of x/Cosine Form}} == Multiple of Sine plus Multiple of Cosine == {{:Multiple of Sine plus Multiple of Cosine}}


Multiple of Sine plus Multiple of Cosine/Cosine Form

:$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \cos {x + \arctan \dfrac {-p} q}$


Multiple of Sine plus Multiple of Cosine/Sine Form

:$p \sin x + q \cos x = \sqrt {p^2 + q^2} \sin \left({x + \arctan \dfrac q p}\right)$


Primitive of Cosine of a x over Sine of a x plus phi

:$\displaystyle \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } = \frac {\ln \size {\map \sin {a x + \phi} } } {a \cos \phi} + \tan \phi \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } + C$


Primitive of Sine of a x over Sine of a x plus phi

:$\displaystyle \int \frac {\sin a x \ \mathrm d x} {\sin \left({a x + \phi}\right)} = \frac x {\cos \phi} - \tan \phi \int \frac {\cos a x \ \mathrm d x} {\sin \left({a x + \phi}\right)} + C$


Reciprocal of One Plus Hyperbolic Cosine

:$\dfrac 1 {1 + \cosh x} = \dfrac 1 2 \operatorname{sech}^2 \dfrac x 2$


Reciprocal of Hyperbolic Cosine Minus One

:$\dfrac 1 {\cosh x - 1} = \dfrac 1 2 \csch^2 \dfrac x 2$


Triangle Right-Angle-Hypotenuse-Side Equality

If two triangles have: : one right angle each : the sides opposite to the right angle equal : another two respective sides equal they will also have: : their third sides equal : the remaining two angles equal to their respective remaining angles.


Internal Angles of Regular Polygon

The size $A$ of each internal angle of a regular $n$-gon is given by: :$A = \dfrac {\paren {n - 2} 180 \degrees} n$


Sum of Internal Angles of Polygon

The sum $S$ of all internal angles of a polygon with $n$ sides is given by the formula $S = \paren {n - 2} 180 \degrees$.


Five Platonic Solids

There exist exactly five platonic solids: :$\paren 1: \quad$ the regular tetrahedron :$\paren 2: \quad$ the cube :$\paren 3: \quad$ the regular octahedron :$\paren 4: \quad$ the regular dodecahedron :$\paren 5: \quad$ the regular icosahedron. {{:Euclid:Proposition/XIII/18/Endnote}}


Geometric Sequence with Coprime Extremes is in Lowest Terms

Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a geometric sequence of integers. Let: :$a_0 \perp a_n$ where $\perp$ denotes coprimality. Then $G_n$ is in its lowest terms.


Measurements of Common Angles/Straight Angle

The measurement of a straight angle is $\dfrac{360^\circ} 2 = 180^\circ$ or $\dfrac {2 \pi} 2 = \pi$.


Measurements of Common Angles/Right Angle

The measurement of a right angle is $\dfrac {180 \degrees} 2 = 90 \degrees$ or $\dfrac \pi 2$.


Measurements of Common Angles/Full Angle

A full angle is equal to one full rotation.


Measurements of Common Angles/Acute Angle

An acute angle measures $\theta$, where: :$0 \degrees < \theta < 90 \degrees$ or: :$0 < \theta < \dfrac \pi 2$


Measurements of Common Angles/Obtuse Angle

An obtuse angle measures $\theta$, where: : $90^\circ < \theta < 180^\circ$ or: : $\dfrac \pi 2 < \theta < \pi$


Measurements of Common Angles/Reflex Angle

A reflex angle measures $\theta$, where: : $180^\circ < \theta < 360^\circ$ or: : $\pi < \theta < 2 \pi$


Supplementary Interior Angles implies Parallel Lines

Given two infinite straight lines which are cut by a transversal, if the interior angles on the same side of the transversal are supplementary, then the lines are parallel.


Equal Corresponding Angles implies Parallel Lines

Given two infinite straight lines which are cut by a transversal, if the corresponding angles are equal, then the lines are parallel.


Equivalence of Definitions of Tangent Vector

Let $M$ be a smooth manifold. Let $m \in M$ be a point.  Let $V$ be an open neighborhood of $m$.  Let $C^\infty \left({V, \R}\right)$ be defined as the set of all smooth mappings $f: V \to \R$. {{TFAE|def = Tangent Vector}}


Parallelism implies Equal Alternate Angles

Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the alternate angles are equal.


Parallelism implies Equal Corresponding Angles

Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the corresponding angles are equal.


Proportion is Reflexive

Proportion is a reflexive relation. That is, every real variable is proportional to itself: :$\forall x \in \R: x \propto x$


Proportion is Symmetric

Proportion is a symmetric relation. That is: :$\forall x, y \in \R: x \propto y \implies y \propto x$


Proportion is Transitive

Proportion is a transitive relation. That is: :$\forall x,y,z \in \R: x \propto y \land y \propto z \implies x \propto z$


Proportion is Equivalence Relation

Proportion is an equivalence relation.


Parallelism implies Supplementary Interior Angles

Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the interior angles on the same side of the transversal are supplementary.


Sine of 18 Degrees

:$\sin 18 \degrees = \sin \dfrac \pi {10} = \dfrac {\sqrt 5 - 1} 4$ where $\sin$ denotes the sine function.


Cosine of 72 Degrees

:$\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = \dfrac {\sqrt 5 - 1} 4$


Sine of 72 Degrees

:$\sin 72 \degrees = \sin \dfrac {2 \pi} 5 = \dfrac {\sqrt{10 + 2 \sqrt 5} } 4$ where $\sin$ denotes the sine function.


Cosine of 18 Degrees

:$\cos 18^\circ = \cos \dfrac \pi {10} = \dfrac {\sqrt{10 + 2 \sqrt 5}} 4$ where $\cos$ denotes the cosine function.


Sine of 3 Degrees

:$\sin 3^\circ = \sin \dfrac \pi {60} = \dfrac {\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}$ where $\sin$ denotes the sine function.


Cosine of 3 Degrees

:$\cos 3^\circ = \cos \dfrac {\pi} {60} = \dfrac{ \sqrt{30} - \sqrt{10} - \sqrt 6 + \sqrt 2 + 2 \sqrt{15 + 3 \sqrt 5} + 2 \sqrt{5 + \sqrt 5} } {16}$ where $\cos$ denotes the cosine function.


Square Root of Sum as Sum of Square Roots

Let $a, b \in \R, a \ge b$. Then: :$\sqrt {a + b} = \sqrt {\dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2}$


Sine of 1 Degree

:$\sin 1^\circ = \sin \dfrac {\pi} {180} = $ where $\sin$ denotes the sine function.


Rectangle is Parallelogram

Let $ABCD$ be a rectangle. Then $ABCD$ is a parallelogram.


Line at Right Angles to Diameter of Circle/Porism


Characteristic Function of Union/Variant 2

:$\chi_{A \mathop \cup B} = \chi_A + \chi_B - \chi_{A \mathop \cap B}$


Characteristic Function of Intersection/Variant 1

:$\chi_{A \cap B} = \chi_A \chi_B$


Sum of Square Roots as Square Root of Sum

:$\sqrt a + \sqrt b = \sqrt {a + b + \sqrt {4 a b} }$


Divisor of One of Coprime Numbers is Coprime to Other

Let $a, b \in \N$ be numbers such that $a$ and $b$ are coprime: :$a \perp b$ Let $c > 1$ be a divisor of $a$: :$c \divides a$ Then $c$ and $b$ are coprime: :$c \perp b$ {{:Euclid:Proposition/VII/23}}


Integer Coprime to all Factors is Coprime to Whole/Proof 1

Let $a, b \in \Z$. Let $\displaystyle b = \prod_{j \mathop = 1}^r b_j$ Let $a$ be coprime to each of $b_1, \ldots, b_r$. Then $a$ is coprime to $b$.


Integer Coprime to all Factors is Coprime to Whole/Proof 2

Let $a, b, c \in \Z$. Let $c$ be coprime to each of $a$ and $b$. Then $c$ is coprime to $a b$. {{:Euclid:Proposition/VII/24}}


Square of Coprime Number is Coprime

Let $a$ and $b$ be coprime integers: :$a, b \in \Z: a \perp b$ Then: :$a^2 \perp b$ {{:Euclid:Proposition/VII/25}}


Product of Coprime Pairs is Coprime

Let $a, b, c, d$ be integers. Let:  :$a \perp c, b \perp c, a \perp d, b \perp d$ where $a \perp c$ denotes that $a$ and $c$ are coprime. Then: :$a b \perp c d$ {{:Euclid:Proposition/VII/26}}


Powers of Coprime Numbers are Coprime

Let $a, b$ be coprime integers: :$a \perp b$ Then: :$\forall n \in \N_{>0}: a^n \perp b^n$ {{:Euclid:Proposition/VII/27}}


Numbers are Coprime iff Sum is Coprime to Both

Let $a, b$ be integers. Then: :$a \perp b \iff a \perp \paren {a + b}$ where $a \perp b$ denotes that $a$ and $b$ are coprime. {{:Euclid:Proposition/VII/28}}


Natural Number is Prime or has Prime Factor

Let $a$ be a natural number greater than $1$. Then either: :$a$ is a prime number or: :there exists a prime number $p \ne a$ such that $p \divides a$ where $\divides$ denotes '''is a divisor of'''. {{:Euclid:Proposition/VII/32}}


Negative of Real Function that Increases Without Bound

Let $f: \R \to \R$ be a real function. Then: :$(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} -\map f x = -\infty$ :$(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} -\map f x = -\infty$


Integer Divided by Divisor is Integer

Let $a, b \in \N$. Then: :$b \divides a \implies \dfrac 1 b \times a \in \N$ where $\divides$ denotes divisibilty. {{:Euclid:Proposition/VII/37}}


Divisor is Reciprocal of Divisor of Integer

Let $a, b, c \in \Z_{>0}$. Then: :$b = \dfrac 1 c \times a \implies c \divides a$ where $\divides$ denotes divisibilty. {{:Euclid:Proposition/VII/38}}


Negative of Real Function that Decreases Without Bound

Let $f: \R \to \R$ be a real function. Then: :$(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} -\map f x = +\infty$ :$(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} -\map f x = +\infty$


Construction of Geometric Sequence in Lowest Terms

It is possible to find a geometric sequence of integers $G_n$ of length $n + 1$ with a given common ratio such that $G_n$ is in its lowest terms. {{:Euclid:Proposition/VIII/2}}


Geometric Sequence in Lowest Terms has Coprime Extremes

A geometric sequence of integers in lowest terms has extremes which are coprime. {{:Euclid:Proposition/VIII/3}}


First Element of Geometric Sequence not dividing Second

Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$. Let $a_0$ not be a divisor of $a_1$. Then: :$\forall j, k \in \set {0, 1, \ldots, n}, j \ne k: a_j \nmid a_k$ That is, if the initial term of $P$ does not divide the second, no term of $P$ divides any other term of $P$. {{:Euclid:Proposition/VIII/6}}


First Element of Geometric Sequence that divides Last also divides Second

Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$. Let $a_0$ be a divisor of $a_n$. Then $a_0$ is a divisor of $a_2$. {{:Euclid:Proposition/VIII/7}}


Geometric Sequences in Proportion have Same Number of Elements

Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$. Let $r$ be the common ratio of $P$. Let $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le m}$ be a geometric sequence of integers of length $m$. Let $r$ be the common ratio of $Q$. Let $b_0$ and $b_m$ be such that $\dfrac {b_0} {b_m} = \dfrac {a_0} {a_n}$. Then $m = n$. {{:Euclid:Proposition/VIII/8}}


Elements of Geometric Sequence between Coprime Numbers

Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n + 1$. Let $a_0$ be coprime to $a_n$. Then there exist geometric sequences of integers $Q_1$ and $Q_2$ of length $n + 1$ such that: :the initial term of both $Q_1$ and $Q_2$ is $1$ :the final term of $Q_1$ is $a_0$ :the final term of $Q_2$ is $a_n$. {{:Euclid:Proposition/VIII/9}}


Product of Geometric Sequences from One

Let $Q_1 = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ and $Q_2 = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ be geometric sequences of integers of length $n + 1$. Let $a_0 = b_0 = 1$. Then the sequence $P = \sequence {p_j}_{0 \mathop \le j \mathop \le n}$ defined as: :$\forall j \in \set {0, \ldots, n}: p_j = b^j a^{n - j}$ is a geometric sequence. {{:Euclid:Proposition/VIII/10}}


Powers of Elements of Geometric Sequence are in Geometric Sequence

Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers. Then the sequence $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ defined as: :$\forall j \in \set {0, 1, \ldots, n}: b_j = a_j^m$ where $m \in \Z_{>0}$, is a geometric sequence. {{:Euclid:Proposition/VIII/13}}


Number divides Number iff Square divides Square

Let $a, b \in \Z$. Then: :$a^2 \divides b^2 \iff a \divides b$ where $\divides$ denotes integer divisibility. {{:Euclid:Proposition/VIII/14}}


Number divides Number iff Cube divides Cube

Let $a, b \in \Z$. Then: :$a^3 \divides b^3 \iff a \divides b$ where $\divides$ denotes integer divisibility. {{:Euclid:Proposition/VIII/15}}


Number does not divide Number iff Cube does not divide Cube

Let $a, b \in \Z$ be integers. Then: :$a \nmid b \iff a^3 \nmid b^3$ where $a \nmid b$ denotes that $a$ is not a divisor of $b$. {{:Euclid:Proposition/VIII/17}}


If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube

Let $P = \tuple {a, b, c, d}$ be a geometric sequence of integers. Let $a$ be a cube number. Then $d$ is also a cube number. {{:Euclid:Proposition/VIII/23}}


If Ratio of Square to Number is as between Two Squares then Number is Square

Let $a, b, c, d \in \Z$ be integers such that: :$\dfrac a b = \dfrac {c^2} {d^2}$ Let $a$ be a square number. Then $b$ is also a square number. {{:Euclid:Proposition/VIII/24}}


Square of Cube Number is Cube

Let $a \in \N$ be a natural number. Let $a$ be a cube number. Then $a^2$ is also a cube number. {{:Euclid:Proposition/IX/3}}


Cube Number multiplied by Cube Number is Cube

Let $a, b \in \N$ be natural numbers. Let $a$ and $b$ be cube numbers. Then $a b$ is also a cube number. {{:Euclid:Proposition/IX/4}}


Number Squared making Cube is itself Cube

Let $a \in \Z$ be an integer. Let $a^2$ be a cube number. Then $a$ is a cube number. {{:Euclid:Proposition/IX/6}}


Product of Composite Number with Number is Solid Number

Let $a, b \in \Z$ be positive integers. Let $a$ be a composite number. Then $a b$ is a solid number. {{:Euclid:Proposition/IX/7}}


Square Numbers are Similar Plane Numbers

Let $a$ and $b$ be square numbers. Then $a$ and $b$ are similar plane numbers.


Elements of Geometric Sequence from One where First Element is Power of Number

Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Let $m \in \Z_{> 0}$. Let $a_1$ be the $m$th power of an integer. Then all the terms of $G_n$ are $m$th powers of  integers. {{:Euclid:Proposition/IX/9}}


Sum of Pair of Elements of Geometric Sequence with Three Elements in Lowest Terms is Coprime to other Element

Let $P = \tuple {a, b, c}$ be a geometric sequence of integers in its lowest terms. Then $\paren {a + b}$, $\paren {b + c}$ and $\paren {a + c}$ are all coprime to each of $a$, $b$ and $c$. {{:Euclid:Proposition/IX/15}}


Two Coprime Integers have no Third Integer Proportional

Let $a, b \in \Z_{>0}$ be integers such that $a$ and $b$ are coprime. Then there is no integer $c \in \Z$ such that: :$\dfrac a b = \dfrac b c$ {{:Euclid:Proposition/IX/16}}


Last Element of Geometric Sequence with Coprime Extremes has no Integer Proportional as First to Second

Let $G_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers such that $a_0 \ne 1$. Let $a_0 \perp a_n$, where $\perp$ denotes coprimality. Then there does not exist an integer $b$ such that: :$\dfrac {a_0} {a_1} = \dfrac {a_n} b$ {{:Euclid:Proposition/IX/17}}


Condition for Existence of Third Number Proportional to Two Numbers

Let $a, b, c \in \Z$ be integers. Let $\tuple {a, b, c}$ be a geometric sequence. In order for this to be possible, both of these conditions must be true: :$(1): \quad a$ and $b$ cannot be coprime :$(2): \quad a \divides b^2$ where $\divides$ denotes divisibility. {{:Euclid:Proposition/IX/18}}


Odd Divisor of Even Number also divides its Half

Let $a, b \in \Z$ be integers. Let $a$ be odd and $b$ be even. Let: :$a \divides b$ where $\divides$ denotes divisibility. Then: :$a \divides \dfrac b 2$  {{:Euclid:Proposition/IX/30}}


Odd Number Coprime to Number is also Coprime to its Double

Let $a, b \in \Z$ be integers. Let $a$ be odd. Let: : $a \perp b$ where $\perp$ denotes coprimality. Then: : $a \perp 2 b$ {{:Euclid:Proposition/IX/31}}


Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd

Let $a \in \Z$ be an integer such that: :$(1): \quad a$ is not a power of $2$ :$(2): \quad \dfrac a 2$ is an even integer. Then $a$ is both even-times even and even-times odd. {{:Euclid:Proposition/IX/34}}


Scaled Real Function that Increases Without Bound

Let $f: \R \to \R$ be a real function. Let $\lambda \in \R_{\ne 0}$ be a nonzero constant. Then: For $\lambda > 0$: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = + \infty$ For $\lambda < 0$: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = -\infty$


Scaled Real Function that Decreases Without Bound

Let $f: \R \to \R$ be a real function. Let $\lambda \in \R_{\ne 0}$ be a nonzero constant. Then: For $\lambda > 0$: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = -\infty$ For $\lambda < 0$: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = +\infty$


Existence of Fraction of Number Smaller than Given

Let $a, b \in \R_{>0}$ be two (strictly) positive real numbers, such that $a > b$. Let the sequence $\left\langle{a_n}\right\rangle$ be defined recursively as: :$a_i = \begin{cases} a & : i = 1 \\ a_{i - 1} - c: \dfrac {a_{i - 1} } 2 < c < a_{i - 1} & : i > 1 \end{cases}$ Then: : $\exists n \in \N_{>0}: a_n < b$ {{:Euclid:Proposition/X/1}}


Restriction of Continuous Mapping is Continuous/Topological Spaces

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $M_1 \subseteq S_1$ be a subset of $S_1$. Let $f: S_1 \to S_2$ be a mapping which is continuous. Let $M_2 \subseteq S_2$ be a subset of $S_2$ such that $f\left[{M_1}\right] \subseteq M_2$. Let $f {\restriction_{M_1 \times M_2}}: M_1 \to M_2$ be the restriction of $f$ to $M_1 \times M_2$. Then $f {\restriction_{M_1 \times M_2}}$ is continuous, where $M_1$ and $M_2$ are equipped with the respective subspace topologies.


Restriction of Continuous Mapping is Continuous/Metric Spaces

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $S \subseteq M_1$ be a subset of $M_1$. Let $f: M_1 \to M_2$ be a mapping which is continuous at a point $\alpha \in S$. Let $f \restriction_S = g: S \to M_2$ be the restriction of $f$ to $S$. Then $g$ is continuous at $\alpha$.


Restriction of Inverse is Inverse of Restriction

Let $S_1$ and $S_2$ be sets. Let $f: S_1 \to S_2$ be a bijection. Let $S \subseteq S_1$ be a subset of $S_1$. Let $f^{-1}$ be the inverse of $f$. Let $f {\restriction_{S \times f \left[{S}\right]}}$ be the restriction of $f$ to $S \times f\left[{S}\right]$. Let $f^{-1} {\restriction_{f \left[{S}\right] \times S}}$ be the restriction of $f^{-1}$ to $f \left[{S}\right] \times S$. Then: : $f {\restriction_{S \times f \left[{S}\right]}}$ is a bijection and: : ${\left({f {\restriction_{S \times f \left[{S}\right]}} }\right)}^{-1} = f^{-1} {\restriction_{f \left[{S}\right] \times S}}$


Hölder's Inequality/General

Let $\struct {X, \Sigma, \mu}$ be a measure space. For $i = 1, \ldots, n$ let $p_i \in \R$ such that: :$\displaystyle \sum_{i \mathop = 1}^n \frac 1 {p_i} = 1$ Let $f_i \in \map {\LL^{p_i} } \mu, f_i: X \to \R$, where $\LL$ denotes Lebesgue space. Then their pointwise product $\displaystyle \prod_{i \mathop = 1}^n f_i$ is integrable, that is: :$\displaystyle \prod_{i \mathop = 1}^n f_i \in \map {\LL^1} \mu$ and: :$\displaystyle \norm {\prod_{i \mathop = 1}^n f_i}_1 = \int \size {\prod_{i \mathop = 1}^n f_i} \rd \mu \le \prod_{i \mathop = 1}^n \norm {f_i}_{p_i}$ where the various instances of $\norm {\, \cdot \,}$ signify $p$-seminorms.


Theorem of Even Perfect Numbers/Sufficient Condition

Let $n \in \N$ be such that $2^n - 1$ is prime. Then $2^{n - 1} \paren {2^n - 1}$ is perfect. {{:Euclid:Proposition/IX/36}}


Theorem of Even Perfect Numbers/Necessary Condition

Let $a \in \N$ be an even perfect number. Then $a$ is in the form: :$2^{n - 1} \paren {2^n - 1}$ where $2^n - 1$ is prime.


Condition for Commensurability of Roots of Quadratic Equation/Lemma


Condition for Incommensurability of Roots of Quadratic Equation

Consider the quadratic equation: :$(1): \quad a x - x^2 = \dfrac {b^2} 4$ Then $x$ and $a - x$ are incommensurable {{iff}} $\sqrt {a^2 - b^2}$ and $a$ are incommensurable. {{:Euclid:Proposition/X/18}}


Product of Rationally Expressible Numbers is Rational/Lemma


Product of Rationally Expressible Numbers is Rational

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be the lengths of rational line segments. Furthermore, let $\dfrac a b \in \Q$. Then, $a b \in \Q$. {{:Euclid:Proposition/X/19}}


Construction of Apotome is Unique

Let $D$ be the domain $\left\{{x \in \R_{>0} : x^2 \in \Q}\right\}$, the rationally expressible numbers. Let $a, b \in D$ be two rationally expressible numbers such that $a - b$ is an apotome. Then, there exists only one $x \in D$ such that $a - b + x$ and $a$ are commensurable in square only. {{:Euclid:Proposition/X/79}}


Lower Bound of Natural Logarithm

:$\forall x \in \R_{>0}: 1 - \dfrac 1 x \le \ln x$ where $\ln x$ denotes the natural logarithm of $x$.


Bounds of Natural Logarithm

Let $\ln y$ be the natural logarithm of $y$ where $y \in \R_{>0}$. Then $\ln$ satisfies the compound inequality: :$\displaystyle 1 - \frac 1 y \le \ln y \le y - 1$


Construction of Solid Angle from Three Plane Angles any Two of which are Greater than Other Angle/Lemma


Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights


Construction of Polyhedron in Outer of Concentric Spheres/Porism


Definition:Elevated/Point

Let $P$ be a plane. An '''elevated point (relative to $P$)''' is a point in space which does not lie on $P$.


Definition:Elevated/Line

Let $P$ be a plane. An '''elevated line (relative to $P$)''' is a line in space which does not lie entirely in $P$.


Common Ratio in Rational Geometric Sequence is Rational

Let $\sequence {a_k}$ be a geometric sequence whose terms are rational. Then the common ratio of $\sequence {a_k}$ is rational.


Common Ratio in Integer Geometric Sequence is Rational

Let $\sequence {a_k}$ be a geometric sequence whose terms are all integers. Then the common ratio of $\sequence {a_k}$ is rational.


Form of Geometric Sequence of Integers

Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n + 1$ consisting of integers only. Then the $j$th term of $P$ is given by: :$a_j = k p^{n - j} q^j$ where: : the common ratio of $P$ expressed in canonical form is $\dfrac q p$ : $k$ is an integer.


Geometric Sequence of Integers with Integer Common Ratio

Let $P = \sequence {a_j}_{1 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n$ consisting entirely of integers. Let $r$ be the common ratio of $P$. Then $r$ is an integer {{iff}}: :$\forall i, j \in \set {1, 2, \ldots, n}, i \le j: a_i \divides a_j$ That is, terms of $P$ divide later terms of $P$ {{iff}} $r$ is an integer.


Integers are Coprime iff Powers are Coprime

Let $a, b \in \Z$ be integers. Then: :$a \perp b \iff \forall n \in \N: a^n \perp b^n$ That is, two integers are coprime {{iff}} all their positive integer powers are coprime.


Integer Addition is Cancellable

The operation of addition on the set of integers $\Z$ is cancellable: :$\forall x, y, z \in \Z: x + z = y + z \implies x = y$


Complement Relative to Subset is Subset of Complement Relative to Superset

Let $A, B, C$ be sets such that $A \subseteq B \subseteq C$. Then: :$\relcomp B A \subseteq \relcomp C A$


Complement of Relative Complement is Union with Complement

Let $A, B, C$ be sets such that $A \subseteq B \subseteq C$. Then: :$\relcomp C {\relcomp B A} = A \cup \relcomp C B$


Set Union Preserves Subsets/Families of Sets

Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$. Let: :$\forall \beta \in I: A_\beta \subseteq B_\beta$ Then: :$\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$


Set Union Preserves Subsets/General Result

Let $\mathbb S, \mathbb T$ be sets of sets. Suppose that for each $S \in \mathbb S$ there exists a $T \in \mathbb T$ such that $S \subseteq T$. Then $\bigcup \mathbb S \subseteq \bigcup \mathbb T$.


Intersection of Family is Subset of Intersection of Subset of Family

Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$. Let $J \subseteq I$. Then: :$\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in J} A_\alpha$ where $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection of $\family {A_\alpha}_{\alpha \mathop \in I}$.


Union of Subset of Family is Subset of Union of Family

Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$. Let $J \subseteq I$ Then: :$\displaystyle \bigcup_{\alpha \mathop \in J} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} A_\alpha$ where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.


Intersection of Open Intervals of Positive Reals is Empty

Let $\R_{> 0}$ be the set of strictly positive real numbers. For all $x \in \R_{> 0}$, let $A_x$ be the open real interval $\openint 0 x$. Then: :$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } A_x = \O$


Union of Closed Intervals of Positive Reals is Set of Positive Reals

Let $\R_{> 0}$ be the set of strictly positive real numbers. For all $x \in \R_{> 0}$, let $B_x$ be the closed real interval $\closedint 0 x$. Then: :$\displaystyle \bigcup_{x \mathop \in \R_{> 0} } B_x = \R_{\ge 0}$


Subset of Cartesian Product not necessarily Cartesian Product of Subsets

Let $A$ and $B$ be sets. Let $A$ and $B$ both have at least two distinct elements. Then there exists $W \subseteq A \times B$ such that $W$ is not the cartesian product of a subset of $A$ and a subset of $B$.


Subset equals Preimage of Image implies Injection

Let $f: S \to T$ be a mapping. Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$. Similarly, let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$. Let: :$\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$ Then $f$ is an injection.


Preimage of Image of Subset under Injection equals Subset

Let $f: S \to T$ be an injection. Then: :$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$ where: :$f \sqbrk A$ denotes the image of $A$ under $f$ :$f^{-1}$ denotes the inverse of $f$ :$f^{-1} \circ f$ denotes composition of $f^{-1}$ and $f$.


Subset equals Image of Preimage implies Surjection

Let $f: S \to T$ be a mapping. Let: :$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$ where $f \sqbrk B$ denotes the image of $B$ under $f$. Then $f$ is a surjection.


Image of Preimage of Subset under Surjection equals Subset

Let $f: S \to T$ be a surjection. Then: :$\forall B \subseteq T: B = \left({f \circ f^{-1} }\right) \sqbrk B$ where: :$f \sqbrk B$ denotes the image of $B$ under $f$ :$f^{-1}$ denotes the inverse of $f$ :$f \circ f^{-1}$ denotes composition of $f$ and $f^{-1}$.


Intersection of Image with Subset of Codomain

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let $A \subseteq S$ and $B \subseteq T$. Then: :$f \sqbrk {A \cap f^{-1} \sqbrk B} = f \sqbrk A \cap B$


Projection is Injection iff Factor is Singleton

Let $S_1, S_2, \ldots, S_n$ be non-empty sets. Let $\displaystyle S = \prod_{i \mathop = 1}^n S_i$ be the cartesian product of $S_1, S_2, \ldots, S_n$. Let $\pr_j: S \to S_j$ be the $j$th projection on $S$. Then $\pr_j$ is an injection {{iff}} $S_k$ is a singleton for all $k \in \set {1, 2, \dotsc, n}$ where $k \ne j$.


Preimage of Element under Projection

Let $A$ and $B$ be sets. Let $A \times B$ be the cartesian product of $A$ and $B$. Let $\pr_1: A \times B \to A$ be the first projection of $A \times B$. Let $a \in A$. Then: :$\pr_1^{-1} \sqbrk {\set a} = \set {\tuple {a, b}: b \in B}$ that is: :$\pr_1^{-1} \sqbrk {\set a} = \set a \times B$


Divisors of Factorial

Let $n \in \N_{>0}$. Then all natural numbers less than or equal to $n$ are divisors of $n!$: :$\forall k \in \left\{{1, 2, \ldots, n}\right\}: n! \equiv 0 \pmod k$


Positive Difference Relation on Reals is Transitive

Let $P \subseteq \R$ be a subset of the real numbers such that: :$(1): \quad 1 \in P$ :$(2): \quad a, b \in P \implies a + b \in P$ :$(3): \quad$ For all $x \in \R$, exactly one of these is true: :::$x \in P$ :::$x = 0$ :::$-x \in P$ Let $Q \subseteq \R \times \R$ be the relation on $\R$ defined as: :$Q = \set {\tuple {a, b} \in \R:  a - b \in P}$ Then $Q$ is a transitive relation.


Composition of Repeated Compositions of Injections

Let $S$ be a set. Let $f: S \to S$ be an injection. Let the sequence of mappings: :$f^0, f^1, f^2, \ldots, f^n, \ldots$ be defined as: :$\forall n \in \N: f^n \left({x}\right) = \begin{cases} x & : n = 0 \\ f \left({x}\right) & : n = 1 \\ f \left({f^{n-1} \left({x}\right)}\right) & : n > 1 \end{cases}$ Then: :$\forall m, n \in \Z_{\ge 0}: f^n \circ f^m = f^{m + n}$ where $f^n \circ f^m$ denotes composition of mappings.


Derivative Function on Set of Functions induces Equivalence Relation

Let $X$ be the set of real functions $f: \R \to \R$ which possess continuous derivatives. Let $\mathcal R \subseteq X \times X$ be the relation on $X$ defined as: :$\mathcal R = \set {\tuple {f, g} \in X \times X: D f = D g}$ where $D f$ denotes the first derivative of $f$. Then $\mathcal R$ is an equivalence relation.


Factorial Divides Product of Successive Numbers

Let $m, n \in \N_{\ge 1}$ be natural numbers Let $m^{\overline n}$ be $m$ to the power of $n$ rising. Then: :$m^{\overline n} \equiv 0 \bmod n!$ That is, the factorial of $n$ divides the product of $n$ successive numbers.


Binomial Coefficient is Integer

Let $\dbinom n k$ be a binomial coefficient. Then $\dbinom n k$ is an integer.


Set of Mappings which map to Same Element induces Equivalence Relation

Let $X$ and $Y$ be sets. Let $E$ be the set of all mappings from $X$ to $Y$. Let $b \in X$. Let $\mathcal R \subseteq E \times E$ be the relation on $E$ defined as: :$\mathcal R := \set {\tuple {f, g} \in \mathcal R: \map f b = \map g b}$ Then $\mathcal R$ is an equivalence relation.


Equivalence of Definitions of Inverse Mapping

Let $S$ and $T$ be sets. {{TFAE|def = Inverse Mapping}}


Inverse Mapping is Bijection

Let $S$ and $T$ be sets. Let $f: S \to T$ and $g: T \to S$ be inverse mappings of each other. Then $f$ and $g$ are bijections.


Mapping is Injection and Surjection iff Inverse is Mapping/Proof 2

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Then: : $f: S \to T$ can be defined as a bijection in the sense that: ::$(1): \quad f$ is an injection ::$(2): \quad f$ is a surjection. {{iff}}: :the inverse $f^{-1}$ of $f$ is such that: ::for each $y \in T$, the preimage $\map {f^{-1} } y$ has exactly one element. :That is, such that $f^{-1} \subseteq T \times S$ is itself a mapping.


Mapping is Extension iff Composite with Inclusion

Let $S$ and $T$ be sets. Let $A \subseteq S$. Let $f: S \to T$ and $g: A \to T$ be mappings. Then $f$ is an extension of $g$ {{iff}}: :$f = g \circ i_A$ where $i_A$ is the inclusion mapping on $A$. This can be illustrated using a commutative diagram as follows: ::$\begin {xy} \xymatrix@L + 2mu@ + 1em {  A \ar[r]^*{i_A}      \ar@{-->}[rd]_*{f = g \circ i_A} &  S \ar[d]^*{g} \\ &  T } \end {xy}$


Cardinality of Set of Restrictions of Mapping

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let the cardinality of $S$ be $n$. Let $F$ be the set of restrictions of $f$ to a subset of $S$. Then there are $2^n$ elements of $F$.


Projection is Surjection/Family of Sets

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets. Let $\displaystyle \prod_{\alpha \mathop \in I} S_\alpha$ be the Cartesian product of $\family {S_\alpha}_{\alpha \mathop \in I}$. Let each of $S_\alpha$ be non-empty. For each $\beta \in I$, let $\displaystyle \pr_\beta: \prod_{\alpha \mathop \in I} S_\alpha \to S_\beta$ be the $\beta$th projection on $\displaystyle S = \prod_{\alpha \mathop \in I} S_\alpha$. Then $\pr_\beta$ is a surjection.


Chebyshev Distance on Real Vector Space is Metric

The Chebyshev distance on $\R^n$: : $\displaystyle \forall x, y \in \R^n: d_\infty \left({x, y}\right):= \max_{i \mathop = 1}^n {\left\vert{x_i - y_i}\right\vert}$ is a metric.


Chebyshev Distance is Metric

Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces. Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$. Let $d_\infty: \mathcal A \times \mathcal A \to \R$ be the Chebyshev distance on $\mathcal A$: :$\displaystyle \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, y_i} }$ where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \mathcal A$. Then $d_\infty$ is a metric,


Set of Successive Numbers contains Unique Multiple

Let $m \in \Z_{\ge 1}$. Then $\set {m, m + 1, \ldots, m + n - 1}$ contains a unique integer that is a multiple of $n$. That is, in any set containing $n$ successive integers, $n$ divides exactly one of those integers.


Taxicab Metric is Topologically Equivalent to Chebyshev Distance on Real Vector Space

For $n \in \N$, let $\R^n$ be a real vector space. Let $d_1$ be the taxicab metric on $\R^n$. Let $d_\infty$ be the Chebyshev distance on $\R^n$. Then :$\forall x, y \in \R^n: d_\infty \left({x, y}\right) \le d_1 \left({x, y}\right) \le n \cdot d_\infty \left({x, y}\right)$ It follows that $d_1$ and $d_\infty$ are Lipschitz equivalent.


Euclidean Metric is Metric

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces. Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$. The Euclidean metric on $\mathcal A$ is a metric.


Euclidean Space is Normed Space

The Euclidean norm is a norm on the Euclidean space $\R^n$.


Number of Multiples less than Given Number

Let $m, n \in \N_{\ge 1}$. The number of multiples of $m$ not greater than $n$ is given by: :$q = \floor {\dfrac n m}$ where $\floor {\cdot}$ denotes the floor function