qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,165,759 | <p>I am solving the following question</p>
<p>$$\int\frac{\sin x}{\sin^{3}x + \cos^{3}x}dx.$$</p>
<p>I have been able to reduce it to the following form by diving numerator and denominator by $\cos^{3}x$ and then substituting $\tan x$ for $t$ and am getting the following equation. Should Iis there any other way use p... | mrnovice | 416,020 | <p>Hint: $\frac{t}{t^{3}+1} = \frac{A}{t+1} + \frac{Bt + C}{t^{2}-t+1}$ where $A$ and $B$ and $C$ are constants to be found</p>
<p>Can you solve it now?</p>
<p>In case you get stuck:</p>
<p>$A = -\frac{1}{3}$
$B = \frac{1}{3}$
$C = \frac{1}{3}$</p>
<p>Then we get $I = \int -\frac{1}{3(t+1)} + \frac{t}{3(t^{2}-t+1)}... |
2,222,215 | <p>Determine whether the difference of the following two series is convergent or not and Prove your answer$$
\sum_{n=1}^\infty \frac{1}{n} $$ and $$\sum_{n=1}^\infty \frac{1}{2n-1} $$</p>
<p>What i tried. I said that the difference of the two series is divergent. My proof is as follows. Find the difference of the tw... | Sarvesh Ravichandran Iyer | 316,409 | <p>A few things need correction in your proof. First, let's compliment you for the idea of trying the contrapositive. Unfortunately, the idea falls flat due to a mistake:</p>
<blockquote>
<p>$\sum_{n=1}^\infty \frac{1}{2n-1}$ is <strong>not</strong> convergent!</p>
</blockquote>
<p>To do the question, we need an ob... |
2,222,215 | <p>Determine whether the difference of the following two series is convergent or not and Prove your answer$$
\sum_{n=1}^\infty \frac{1}{n} $$ and $$\sum_{n=1}^\infty \frac{1}{2n-1} $$</p>
<p>What i tried. I said that the difference of the two series is divergent. My proof is as follows. Find the difference of the tw... | Laars Helenius | 112,790 | <p>HINT: Observe that $$\sum_{n=1}^\infty\frac{n-1}{n(2n-1)}\ge\sum_{n=1}^\infty\frac{n-1}{n(2n)}=\sum_{n=1}^\infty\frac{1}{2n}-\sum_{n=1}^\infty\frac{1}{2n^2}.$$</p>
|
2,555,815 | <p><strong>Problem</strong></p>
<p>Let $a_{0}(n) = \frac{2n-1}{2n}$ and $a_{k+1}(n) = \frac{a_{k}(n)}{a_{k}(n+2^k)}$ for $k \geq 0.$</p>
<p>The first several terms in the series $a_k(1)$ for $k \geq 0$ are:</p>
<p>$$\frac{1}{2}, \, \frac{1/2}{3/4}, \, \frac{\frac{1}{2}/\frac{3}{4}}{\frac{5}{6}/\frac{7}{8}}, \, \frac... | mercio | 17,445 | <p>Let $f_0(z) = z$ and $f_{n+1}(z) = f_n(z) / f_n(z+2^n)$</p>
<p>One can show that when $z \to \infty$, the rational fractions $f_n$ for $n \ge 1$ have asymptotic developments at infinity that converge for $|z| > n$, such that
$f_n(z) = 1 + O(z^{-n})$ and $f_n'(z) = O(z^{-n-1})$</p>
<p>Call $s(k) = +1,-1,-1,+1,\... |
1,768,317 | <p>Show that $\sin(x) > \ln(x+1)$ when $x \in (0,1)$. </p>
<p>I'm expected to use the maclaurin series (taylor series when a=0)</p>
<p>So if i understand it correctly I need to show that: </p>
<p>$$\sin(x) = \lim\limits_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{(-1)^{k-1}}{(2k-1)!} \cdot x^{2k-1} > \lim\limi... | Hagen von Eitzen | 39,174 | <p>As the series are alternating with summands strictly decreasing in absolute value (at least for $0<x<1$), we have
$$ \sin x>x-\frac16x^3$$
and
$$ \ln(1+x)<x-\frac12x^2+\frac13x^3$$
Hence the difference is
$$ \sin x-\ln(1+x)>\frac12x^2-\frac12x^3=\frac12x^2(1-x)>0.$$</p>
|
569,103 | <blockquote>
<blockquote>
<p>How can I calculate the first partial derivative $P_{x_i}$ and the second partial derivative $P_{x_i x_i}$ of function:
$$
P(x,y):=\frac{1-\Vert x\rVert^2}{\Vert x-y\rVert^n}, x\in B_1(0)\subset\mathbb{R}^n,y\in S_1(0)?
$$</p>
</blockquote>
</blockquote>
<p>I ask this with reg... | Michael | 155,065 | <p>Here is my work for the first limit: Let $f(x) = \frac{num(x)}{den(x)}$. We get $\infty/\infty$ so we can use L'Hopital: </p>
<p>\begin{align*}
\frac{num'(x)}{den'(x)}&=\frac{3(1+\sec(x))\log \sec x}{(\tan(x))(x + \log(\sec x + \tan x)) + (\log \sec (x)) (1 + \sec(x)) }\\
&= \frac{3(1+\sec(x))\log \sec x... |
287,597 | <p>Can anyone explain how why <a href="http://en.wikipedia.org/wiki/Quaternion#Matrix_representations">the matrix representation of the quaternions using real matrices</a> is constructed as such?</p>
| Thomas | 26,188 | <p>I am not sure exactly what you mean by asking "... why the ...". "Why" questions can be hard to answer satisfactory in math.</p>
<p>The claim is that the Quaternions $\mathbb{H}$ are isomorphic (as $\mathbb{R}$-algebras) to the given set of matrices. The isomorphism looks like this:</p>
<p>$$
\phi: a + bi + cj + d... |
287,597 | <p>Can anyone explain how why <a href="http://en.wikipedia.org/wiki/Quaternion#Matrix_representations">the matrix representation of the quaternions using real matrices</a> is constructed as such?</p>
| G Cab | 317,234 | <p>It is a simple consequence of the multiplication rules of the quaternions. </p>
<p>So
$$
\eqalign{
& \left( {a + bi + cj + dk} \right) \cdot 1 = \left( {\matrix{
a & b & c & d \cr
{ - b} & a & { - d} & c \cr
{ - c} & d & a & { - b} \cr
{ - d} & { - c... |
2,027,044 | <p>Prove:
$$
(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}, \qquad \forall n \in \mathbb{N}
$$
I have have tried using the triangle inequlity $ |a + b| \le |a| + |b| $, without any success.</p>
| Bart Michels | 43,288 | <p><strong>Hint:</strong> (Assuming $a,b$ positive) Binomial theorem.</p>
<blockquote class="spoiler">
<p> let $c=a^{1/n}$, $d=b^{1/n}$. We want $$c^n+d^n\leq(c+d)^n.$$ By the <a href="https://en.wikipedia.org/wiki/Binomial_theorem" rel="nofollow noreferrer">binomial theorem</a>, the RHS contains the LHS plus more p... |
348,748 | <p>Find the solution for $Ax=0$ for the following $3 \times 3$ matrix:</p>
<p>$$\begin{pmatrix}3 & 2& -3\\ 2& -1&1 \\ 1& 1& 1\end{pmatrix}$$</p>
<p>I found the row reduced form of that matrix, which was </p>
<p>$$\begin{pmatrix}1 & 2/3& -1\\ 0& 1&-9/7 \\ 0& 0& 1\end{pm... | Pedro | 23,350 | <p>You have shown that matrix is non singular (or invertible), so that the only solution, as you state, is ${\bf x}=0$.</p>
|
2,464,649 | <p>Find all primes $p$ such that $p+1$ is a perfect square.</p>
<p>All primes except for 2 (3 is not a perfect square, so we can exclude that case) are odd, so we can express them as $2n+1$ for some $n\in\mathbb{Z}_{+}$. Let's express the perfect square as $a^2$, where $a\in\mathbb{Z}_{+}$. Since we are interested in ... | velut luna | 139,981 | <p>If a prime $p$ is of the form of $n^2-1$, then</p>
<p>$$p=(n+1)(n-1)$$</p>
<p>and so $n-1$ must be 1.</p>
|
2,464,649 | <p>Find all primes $p$ such that $p+1$ is a perfect square.</p>
<p>All primes except for 2 (3 is not a perfect square, so we can exclude that case) are odd, so we can express them as $2n+1$ for some $n\in\mathbb{Z}_{+}$. Let's express the perfect square as $a^2$, where $a\in\mathbb{Z}_{+}$. Since we are interested in ... | Community | -1 | <p>I believe that if we have $p+1=k^2$ we can see $p=(k-1)(k+1)$ and hence $3$ is the only such prime.</p>
|
2,464,649 | <p>Find all primes $p$ such that $p+1$ is a perfect square.</p>
<p>All primes except for 2 (3 is not a perfect square, so we can exclude that case) are odd, so we can express them as $2n+1$ for some $n\in\mathbb{Z}_{+}$. Let's express the perfect square as $a^2$, where $a\in\mathbb{Z}_{+}$. Since we are interested in ... | Deniz Tuna Yalçın | 471,430 | <p>If our prime $p$ is in form of $n^2-1,\quad n\in\mathbb{Z}$ then it could also be written as $p=(n-1)(n+1)$ If $p$ is a prime than it cannot be written as the product of two other primes (or the productof any integer other than $1$ and itself) (states the fundamental theorem of arithmetic) If $n-1=1$ then $n=2$ and ... |
24,055 | <p>Running this code:</p>
<pre><code>Histogram[{RandomVariate[NormalDistribution[1/4,0.12],100],
RandomVariate[NormalDistribution[3/4, 0.12], 100]},
Automatic, "Probability", PlotRange -> {{0, 1}, {0, 1}},
Frame -> True, PlotRangeClipping -> True,
FrameLabel -> {Style["x axis", 15], Style["probability... | Silvia | 17 | <p>This issue happened for me in 9.0.1 and also some earlier versions.</p>
<p>My crude yet working workaround is adding some spaces after <strong><em>y</em></strong> to force it been displayed entirely, meanwhile also add corresponding spaces before <strong><em>p</em></strong> to keep the word being centrally aligned.... |
1,242,001 | <p>The following is the notation for Fermat's Last Theorem </p>
<p>$\neg\exists_{\{a,b,c,n\},(a,b,c,n)\in(\mathbb{Z}^+)\color{blue}{^4}\land n>2\land abc\neq 0}a^n+b^n=c^n$ </p>
<p>I understand everything in the notation besides the 4 highlighted in blue. Can someone explain to me what this means?</p>
| Gregory Grant | 217,398 | <p>The only $x$ that makes that zero is $x=-2$. Divide both sides by the $e^{\dots}$ part which you know is never zero.</p>
|
723,570 | <p>In the proof of Theorem 6.11, $\varphi$ is uniformly continuous and hence for arbitrary $\epsilon > 0$ we can pick $\delta > 0$ s.t. $\left|s-t\right| \leq \delta$ implies $\left|\varphi\left(s\right)-\varphi\left(t\right)\right|<\epsilon$. However, I do not understand why he claims that $\delta < \epsil... | Cameron Williams | 22,551 | <p>It should be clear to figure out how many even numbers there are between $100$ and $400$. If you can't see it immediately, consider the amount of even numbers between $0$ and, say, $20$ and draw some conclusions from that.</p>
<p>Then note that $400-100=300$.. This should give a good indicator of the number of numb... |
1,640,383 | <p>We have function $f:\mathbb{R}\rightarrow \mathbb{R}$ with $$f\left(x\right)=\frac{1}{3x+1}\:$$ $$x\in \left(-\frac{1}{3},\infty \right)$$
Write the Maclaurin series for this function.</p>
<p>Alright so from what I learned in class, the Maclaurin series is basically the Taylor series for when we have $x_o=0$ and we... | Aaron Maroja | 143,413 | <p><strong>Hint:</strong> Remember that </p>
<p>$$\frac{1}{1 + x} = \sum_{n=0}^{\infty} (-1)^n x^n \,\,\,\, \text{for} \,\,\,\ |x| <1$$</p>
<p>Then $$\frac{1}{1 +3x} = \ldots$$</p>
<p>for $|3x| < 1 \implies |x| < \frac{1}{3}$</p>
|
242,097 | <p>I need to show that the function $f(n) = n^2$ is not of $\mathcal{O}(n)$. If I am correct I should prove that there is no number $c,n \geq 0$ where $n^2\lt cn$. How to do that?</p>
| Community | -1 | <p>I assume that you want to show that $n^2 \notin \mathcal{O}(n)$. By definition, if $f(n) \in \mathcal{O}(n)$, then there exists $c>0$ such that for all $n > n_0$ we have
$$f(n) \leq c n$$
If there were to exist constants $n_0$ and $c$ such that $$n^2 \leq cn$$ what happens for $n > \max \left\{n_0, c \right... |
233,915 | <p>a,b are elements of the group G</p>
<p>I have no idea how to even start - I was thinking of defining a,b as two square matrices and using the non-commutative property of matrix multiplication but I'm not sure if that's the way to go...</p>
| André Nicolas | 6,312 | <p>Consider the multiplicative subgroup of the <a href="http://en.wikipedia.org/wiki/Quaternion" rel="nofollow">quaternions</a> consisting of $\pm 1$, $\pm i$, $\pm j$, and $\pm k$. Let $a=i$ and $b=j$. We have $ij^2=-i$ and $j^2i=-i$ but $ij\ne ji$.</p>
|
1,289,626 | <blockquote>
<p>find the Range of $f(x) = |x-6|+x^2-1$</p>
</blockquote>
<p>$$ f(x) = |x-6|+x^2-1 =\left\{
\begin{array}{c}
x^2+x-7,& x>0 .....(b) \\
5,& x=0 .....(a) \\
x^2-x+5,& x<0 ......(c)
\end{array}
\right.
$$</p>
<p>from eq (b) i got $$f(x)= \left(x+\frac12\right)^2-\frac{29}4 \ge-\fra... | egreg | 62,967 | <p>No derivatives are necessary. For $x\ge6$ the function is
$$
f(x)=x^2+x-7
$$
The graph is an arc of a parabola with its axis at $x=-1/2$, so in this interval the function is increasing, with its minimum at $6$: $f(6)=35$.</p>
<p>For $x<6$ the function is
$$
f(x)=x^2-x+5
$$
whose graph is an arc of a parabola wit... |
3,375,181 | <p>How do I graph f(x)=1/(1+e^(1/x)) except for replacing variable x with numbers?
Besides, I get the picture of the answer online <a href="https://i.stack.imgur.com/gZb7X.png" rel="nofollow noreferrer">enter image description here</a> and do not understand why x = 0 exists on this graph.</p>
| Certainly not a dog | 691,550 | <p>Noting the form of the LHS, it is instructive to choose the following path:</p>
<p><span class="math-container">$${yy’’-2{y’}^2\over y^2} = 1$$</span>
<span class="math-container">$$\frac{d\frac{y’}y}{dx} = 1+{\biggl(\frac {y’}y\biggr)}^2$$</span>
<span class="math-container">$$x+c = \arctan{\frac{y’}y}$$</span>
<s... |
902,313 | <p>The wikipedia page on clopen sets says "Any clopen set is a union of (possibly infinitely many) connected components." </p>
<p>I thought any topological space is the union of its connected components? Why is this singled out here for clopen sets?</p>
<p>Does it have something to do with it $x\in C$ a clopen subset... | Jamie Walton | 99,825 | <p>Take the space $\mathbb{R}$. Then the subset $\mathbb{R}_{>0}$ of positive elements is a subset of $\mathbb{R}$, but is not a union of connected components of $\mathbb{R}$ (the only connected component of $\mathbb{R}$ is $\mathbb{R}$ itself). But then that's okay, because $\mathbb{R}_{>0}$ isn't clopen in $\ma... |
407,890 | <p>Here comes a second sophisticated version of my conjecture, as critics came up the <a href="https://math.stackexchange.com/questions/407812/conjecture-on-combinate-of-positive-integers-in-terms-of-primes">first version</a> was trivial.</p>
<p>Teorem <a href="https://oeis.org/A226233" rel="nofollow noreferrer">2</a>... | Jyrki Lahtonen | 11,619 | <p>If I have understood the question correctly, the claim can be seen to be correct as follows. </p>
<p>The definition of the integer $b_u$ tells us that $b_u$ is the unique integer in the range $1\le b_u\le p-1$ that is congruent to $u$ modulo $p-1$. So choosing $u$ from the correct residue class modulo $p-1$ allows ... |
197,441 | <p>I have a list,</p>
<pre><code>l1 = {{a, b, 3, c}, {e, f, 5, k}, {n, k, 12, m}, {s, t, 1, y}}
</code></pre>
<p>and want to apply differences on the third parts and keep the parts right of the numerals collected.</p>
<p>My result should be</p>
<pre><code>l2 = {{2, c, k}, {7, k, m}, {-11, m, y}}
</code></pre>
<p>I... | kglr | 125 | <p>You can also use <a href="https://reference.wolfram.com/language/ref/BlockMap.html" rel="nofollow noreferrer"><code>BlockMap</code></a> as follows:</p>
<pre><code>BlockMap[{#[[3]].{-1, 1}, ## & @@ Flatten@#[[4 ;;]]} &@* Transpose, l1, 2, 1]
</code></pre>
<blockquote>
<p>{{2, c, k}, {7, k, m}, {-11, m, y}... |
649,495 | <p>Hello I'm having trouble understanding the factorizing of a polynomial as </p>
<p>$$x^4-4x$$</p>
<p>After that, I turned it into $$x(x^3-8)$$</p>
<p>But I don't quite understand how it's factored (the process) as </p>
<p>$$x(x−2)(x^2+2x+4)$$</p>
<p>Thanks!</p>
| taue2pi | 112,397 | <p>You can learn "difference of cubes" $$(x^3-y^3)=(x-y)(x^2+xy+y^2)$$ or the general form $$(x^n-y^n)=(x-y)(x^{n-1}y^0+x^{n-2}y^1+...+x^1y^{n-2}+x^0y^{n-1})$$</p>
|
1,080,858 | <p>Why do we have</p>
<ul>
<li>$u_n=\dfrac{1}{\sqrt{n^2-1}}-\dfrac{1}{\sqrt{n^2+1}}=O\left(\dfrac{1}{n^3}\right)$</li>
<li>$u_n=e-\left(1+\frac{1}{n}\right)^n\sim \dfrac{e}{2n}$</li>
</ul>
<p>any help would be appreciated</p>
| DeepSea | 101,504 | <p><strong>Hint:</strong> $\left(\dfrac{1}{n^2-1}\right)^{1/2} = \left(n^2-1\right)^{-1/2} = \dfrac{1}{n}\left(1-\frac{1}{n^2}\right)^{-1/2} = \dfrac{1}{n} + \mathcal{O}\left(\frac{1}{n^3}\right)$</p>
|
1,080,858 | <p>Why do we have</p>
<ul>
<li>$u_n=\dfrac{1}{\sqrt{n^2-1}}-\dfrac{1}{\sqrt{n^2+1}}=O\left(\dfrac{1}{n^3}\right)$</li>
<li>$u_n=e-\left(1+\frac{1}{n}\right)^n\sim \dfrac{e}{2n}$</li>
</ul>
<p>any help would be appreciated</p>
| RE60K | 67,609 | <p>Using Binomial Exapnsion:</p>
<blockquote>
<p>$$u_n=\dfrac{1}{\sqrt{n^2-1}}-\dfrac{1}{\sqrt{n^2+1}}\\
=(n^2-1)^{-1/2}-(n^2+1)^{-1/2}\\
=h[(1-h^2)^{-1/2}-(1+h^2)^{-1/2}]\quad h:=n^{-1}\\
=h[(1+h^2/2+...)-(1-h^2/2+...)]\\
=h[h^2+...]\\
=O(h^3)=O(n^{-3})$$</p>
</blockquote>
<p>And using the $e$ and $\ln$-Series:</p... |
1,080,858 | <p>Why do we have</p>
<ul>
<li>$u_n=\dfrac{1}{\sqrt{n^2-1}}-\dfrac{1}{\sqrt{n^2+1}}=O\left(\dfrac{1}{n^3}\right)$</li>
<li>$u_n=e-\left(1+\frac{1}{n}\right)^n\sim \dfrac{e}{2n}$</li>
</ul>
<p>any help would be appreciated</p>
| Clement C. | 75,808 | <p>For the first one:
$$\begin{align}
\frac{1}{\sqrt{n^2-1}}-\frac{1}{\sqrt{n^2+1}} &= \frac{1}{n}\left(\frac{1}{\sqrt{1-\frac{1}{n^2}}}-\frac{1}{\sqrt{1+\frac{1}{n^2}}}\right) \\
&= \frac{1}{n}\left(\frac{1}{1-\frac{1}{2n^2}+o\!\left(\frac{1}{n^2}\right)}-\frac{1}{1+\frac{1}{2n^2}+o\!\left(\frac{1}{n^2}\right)... |
702,804 | <p>I just need a sanity check, been thinking about this all morning.</p>
<p>If we use the Mean Value Theorem on a function over the infinite interval (suppose the function's domain is unbounded), i.e.</p>
<p>$$M=\lim\limits_{T \to \infty} \dfrac{1}{2T}\int_{-T}^{T} \text{dt} f(t)$$</p>
<p>There is no way that M can ... | John Gowers | 26,267 | <p>The binomial series expansion that you used - </p>
<p>$$
\frac1{1-y}=\sum_{n=0}^{\infty} y^n
$$</p>
<p>is valid only when $|y|<1$. Otherwise, the series diverges. In your solution, you are substituting in $(x-1)$ for $y$, so your solution is valid for $|x-1|<1$, so your power series is in some sense centre... |
25,137 | <p>I want to find an intuitive analogy to explain how binary addition (more precise: an adder circuit in a computer) works. The point here is to explain the abstract process of <em>adding</em> something by comparing it to something that isn't abstract itself.</p>
<p>In principle: An everyday object or an action that is... | Steven Gubkin | 117 | <p>Why do you need an analogy?</p>
<p>I think just having a bunch of beans to count, and grouping the beans into groups of size 1, 2, 4, 8, 16, etc is intuitive enough without needing an analogy. The comparison with base ten arithmetic, where we group into 1, 10, 100, 1000, etc is clear.</p>
<p>Addition is straightfor... |
3,325,340 | <p>Show that <span class="math-container">$$ \lim\limits_{(x,y)\to(0,0)}\dfrac{x^2y^2}{x^2+y^2}=0$$</span>
My try:
We know that, <span class="math-container">$$ x^2\leq x^2+y^2 \implies x^2y^2\leq (x^2+y^2)y^2 \implies x^2y^2\leq (x^2+y^2)^2$$</span>
Then, <span class="math-container">$$\dfrac{x^2y^2}{x^2+y^2}\leq x^2+... | user0102 | 322,814 | <p><strong>HINT</strong>
<span class="math-container">\begin{align*}
0\leq x^{2} \leq x^{2} + y^{2} \Longleftrightarrow 0\leq \frac{x^{2}}{x^{2}+y^{2}} \leq 1 \Longleftrightarrow 0\leq \frac{x^{2}y^{2}}{x^{2}+y^{2}}\leq y^{2}
\end{align*}</span></p>
<p>Then apply the squeeze theorem.</p>
|
3,325,340 | <p>Show that <span class="math-container">$$ \lim\limits_{(x,y)\to(0,0)}\dfrac{x^2y^2}{x^2+y^2}=0$$</span>
My try:
We know that, <span class="math-container">$$ x^2\leq x^2+y^2 \implies x^2y^2\leq (x^2+y^2)y^2 \implies x^2y^2\leq (x^2+y^2)^2$$</span>
Then, <span class="math-container">$$\dfrac{x^2y^2}{x^2+y^2}\leq x^2+... | MafPrivate | 695,001 | <p><strong>Tips</strong></p>
<p><span class="math-container">$\lim\limits_{\left(x,y\right)\rightarrow\left(0,0\right)} \dfrac{x^2 y^2}{x^2+y^2} = \lim\limits_{\left(x,y\right)\rightarrow\left(0,0\right)} \dfrac{1}{\frac{1}{x^2}+\frac{1}{y^2}}$</span></p>
|
501,660 | <p>In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?</p>
<p>Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.</p>... | Eric Stucky | 31,888 | <p>Congratulations! You've stumbled in to a very interesting question!</p>
<p>In higher mathematics, we often notice that some things which are really easy to talk about but difficult to express rigorously have a property which is really easy to express rigorously but something that we probably wouldn't have thought o... |
501,660 | <p>In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?</p>
<p>Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.</p>... | Michael Hardy | 11,667 | <p>Long before there were power series, in the second century A.D., Ptolemy, a man who wrote in Greek and probably lived in Alexandria, created a table of values of what amounts to the sine function.</p>
<p>See <a href="https://en.wikipedia.org/wiki/Ptolemy%27s_table_of_chords" rel="nofollow noreferrer">this page</a>.<... |
2,791,204 | <p>I am trying to understand whether or not the product of two positive semidefinite matrices is also positive semidefinite. This topic has already been discussed in the past <a href="https://math.stackexchange.com/q/113859">here</a>. For me $A$ is positive definite" means $x^T A x > 0$ for all nonzero real vectors ... | Riccardo Sven Risuleo | 419,249 | <p>The problem is not in $A$ and $B$, but in their product $AB$: the product is not symmetric; hence, there is no clear definition of <em>positive definiteness</em>.</p>
<p>In standard parlance, a <em>Hermitian</em> (or <em>symmetric</em>) matrix $M$ is <em>positive definite</em> if $x^T M x > 0$ for all $x$ (and t... |
153,217 | <blockquote>
<p>Let $$f(x)=\frac{2x+1}{\sin(x)}$$ Find $f'(x).$ </p>
</blockquote>
<p>I used Quotient Rule <br>
$$\begin {align*}\frac{\sin(x)2-(2x+1)\cos(x)}{\sin^2(x)}\\
=\frac{3-2x\cos(x)}{\sin(x)} \end {align*}$$</p>
<p>Is that right? I don't know how to get the answer.
Please help me out, thanks.</p>
| DonAntonio | 31,254 | <p>You had (or should) $$\frac{2\sin x-2x\cos x-\cos x}{\sin^2x}$$ and this can't possibly equal what you wrote (where does that $\,3\,$ come from?)</p>
|
625,821 | <p>$$\int^\infty_0\frac{1}{x^3+1}\,dx$$</p>
<p>The answer is $\frac{2\pi}{3\sqrt{3}}$.</p>
<p>How can I evaluate this integral?</p>
| lab bhattacharjee | 33,337 | <p>Using <a href="http://mathworld.wolfram.com/PartialFractionDecomposition.html" rel="nofollow">Partial Fraction Decomposition</a>,</p>
<p>$$\frac1{1+x^3}=\frac A{1+x}+\frac{Bx+C}{1-x+x^2}$$</p>
<p>Multiply either sides by $(1+x)(1-x+x^2)$ and compare the coefficients of the different powers of $x$ to find $A,B,C$<... |
1,986,247 | <p>The nth Catalan number is :
$$C_n = \frac {1} {n+1} \times {2n \choose n}$$
The problem 12-4 of CLRS asks to find :
$$C_n = \frac {4^n} { \sqrt {\pi} n^{3/2}} (1+ O(1/n)) $$
And Stirling's approximation is:
$$n! = \sqrt {2 \pi n} {\left( \frac {n}{e} \right)}^{n} {\left( 1+ \Theta \left(\frac {1} {n}\right) \right... | Micah | 30,836 | <p>Use the binomial approximation for $(1+y)^k$:</p>
<p>$$
(1+y)^k=1+ky+\Theta(y^2)
$$
as $y \to 0$.</p>
<p>In your case, you can take $k=-1$ to show that any function which is $\frac{1}{1+\Theta(1/n)}$ is also $1+\Theta(\frac{1}{n})$.</p>
|
1,611,560 | <p>Reading through the first half of Baby Rudin again before taking an Analysis class, I came across the assertion that "it is also easy to show that k-cells are convex". </p>
<p>Previously it gave the example of open/closed balls being convex, and the proof is obvious and easy to understand. That being said, and it... | Myo Nyunt | 828,003 | <p>If <span class="math-container">$x,y$</span> are two points of a k-cell, <span class="math-container">$C=\{(x_1,…,x_k )| a_i≤x_i≤b_i \}$</span>, then for <span class="math-container">$1≤i≤k$</span>, <span class="math-container">$a_i≤x_i≤b_i , a_i\le y_i≤b_i$</span> and <span class="math-container">$ λa_i+(1-λ) a_i\l... |
2,293,147 | <p>I was trying to solve this ODE $\frac{dy}{dx} = c_{1} + c_{2}y + \frac{c_{3}}{y} , y(0) = c , c >0$.</p>
<p>where $c_{1},c_{2},c_{3}$ are three real numbers say $c_{1} < 0,c_{2},c_{3} > 0$.</p>
<p>I thought of using separation of variables giving me $x = \int(\frac{y}{c_{1}y+c_{2}y^2+c_{3}})dy + c$.</p>
... | Jaideep Khare | 421,580 | <p>Instead of doing that, why don't you just multiply the whole expression by $y$ and the let $y^2=t$.</p>
|
1,363,144 | <p>Given a cubic polynomial $f(x) = ax^{3} + bx^{2} + cx +d$ with arbitrary real coefficients and $a\neq 0$. Is there an easy test to determine when all the real roots of $f$ are negative?</p>
<p>The Routh-Hurwitz Criterion gives a condition for roots lying in the open left half-plane for an arbitrary polynomial with ... | Daniel Fischer | 83,702 | <p>We can assume that $a = 1$, since dividing by $a$ doesn't change the zeros. Then we know that $\lim\limits_{x\to +\infty} f(x) = +\infty$. We want to check whether $f$ has a zero in $[0,+\infty)$.</p>
<p>If $d = 0$, we have $f(0) = 0$. There can be circumstances when that should count as negative. Then we are reduc... |
1,289,868 | <p>EDIT (<em>now asking how to write $F$ as distributions, instead of writing the integral in terms of distributions</em>): </p>
<p>Let $F$ be the distribution defined by its action on a test function $\phi$ as </p>
<p>\begin{equation*}
F(\phi)=\int_{\pi}^{2\pi}x\phi(x)dx.
\end{equation*}</p>
<p>How would you write... | paul garrett | 12,291 | <p>Although I think @NikitaEvseev's answer is the most natural, perhaps the small exercise of rewriting integration over an interval $[a,b]$ as a distribution is worthwhile. That is, consider
$$
u(\varphi) \;=\; \int_a^b \varphi(x)\;dx
$$
Edit: simpler than what I wrote before:
$$
u(\varphi) \;=\; \int_a^b \varphi(x)\,... |
481,421 | <p>Find the limit of:
$$\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$$</p>
| njguliyev | 90,209 | <p>$$\lim_{x \to \infty} \frac{\cos \frac1x - 1}{\cos \frac2x - 1} = \lim_{x \to \infty} \frac{-2\sin^2 \frac{1}{2x}}{-2\sin^2 \frac{1}{x}} = \lim_{x \to \infty} \frac{\sin^2 \frac{1}{2x}}{\sin^2 \frac{1}{x}} = \lim_{x \to \infty} \frac14 \frac{\sin^2 \frac{1}{2x}}{\frac{1}{(2x)^2}} \frac{\frac{1}{x^2}}{\sin^2 \frac{1}... |
481,421 | <p>Find the limit of:
$$\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$$</p>
| minar | 86,791 | <p>Another option, let $u=exp(i/2x).$ Then, $\cos(1/x)-1=(u^2+1/u^2)/2-1=(u-1/u)^2/2$ and $\cos(2/x)-1=(u^4+1/u^4)/2-1=(u^2-1/u^2)^2/2$.</p>
<p>Thus:
$$\frac{\cos(1/x)-1}{\cos(2/x)-1}
=\left(\frac{u-1/u}{u^2-1/u^2}\right)^2
=\left(\frac{1}{u+1/u}\right)^2.
$$</p>
<p>Since $u$ tends to $1$ as $x$ goes to infinity, you... |
2,195,739 | <p>For
$$
f(x) = \begin{cases}
x^2 & \text{if $x\in\mathbb{Q}$,} \\[4px]
x^3 & \text{if $x\notin\mathbb{Q}$}
\end{cases}
$$</p>
<p>What I did was examine each of the limits at $0$ of
$\displaystyle\lim_{x\to0} \frac{f(x)-f(a)}{x-a}$ for each case but I am not sure </p>
| egreg | 62,967 | <p>You need to compute
$$
\lim_{x\to0}\frac{f(x)-f(0)}{x-0}=\lim_{x\to0}\frac{f(x)}{x}
$$
But you can write, for $x\ne0$,
$$
\frac{f(x)}{x}=\begin{cases}
x & \text{if $x\in\mathbb{Q}$}\\[4px]
x^2 & \text{if $x\notin\mathbb{Q}$}
\end{cases}
$$
For $0<|x|<1$, you have $\left|\dfrac{f(x)}{x}\right|\le |x|$. ... |
2,195,739 | <p>For
$$
f(x) = \begin{cases}
x^2 & \text{if $x\in\mathbb{Q}$,} \\[4px]
x^3 & \text{if $x\notin\mathbb{Q}$}
\end{cases}
$$</p>
<p>What I did was examine each of the limits at $0$ of
$\displaystyle\lim_{x\to0} \frac{f(x)-f(a)}{x-a}$ for each case but I am not sure </p>
| complexCreature | 424,614 | <p>Maybe you prefer definition. You must prove that: <br>
$(\forall \epsilon\gt0)(\exists\delta\gt0) (|x-0|\lt\delta \Rightarrow \frac{|f(x)-f(0)|}{|x-0|}\lt\epsilon$) , that is: <br>
$(\forall \epsilon\gt0)(\exists\delta\gt0) (|x|\lt\delta \Rightarrow \frac{|f(x)|}{|x|}\lt\epsilon$)<br>
Since we are interested only in... |
82,716 | <p>There seems to be two competing(?) formalisms for specifying theories: <a href="http://ncatlab.org/nlab/show/sketch" rel="noreferrer">sketches</a> (as developped by Ehresmann and students, and expanded upon by Barr and Wells in, for example, <a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf" rel="nore... | Zinovy Diskin | 19,786 | <p>So, saying that sketches compete with institutions is not correct because the former is an instance of the latter. The actual content of the question is probably this.</p>
<p>There are two types/styles/paradigms of predicate logic: elementwise (eg, ordinary FOL) and sortwise, or categorical, logic. In the latter, p... |
4,646,498 | <p>I have an assignment for university and I’m a bit confused as to how I should translate the following sentence:</p>
<p>Neither Ana nor Bob can do every exercise but each can do some.</p>
<p>I've identified the atomic sentences A=Ana can do every exercise and B=Bob can do every exercise and managed to translate the f... | A. P. | 1,027,216 | <p>Using reduction of order should be: <span class="math-container">$p=y', p'=y''$</span> and then <span class="math-container">$y''+4x=0$</span> can be written as <span class="math-container">$p'+4x=0$</span> which is first order and then integrating give <span class="math-container">$p=a-2x^2$</span>. Thus, substitut... |
2,573,458 | <p>Given $n$ prime numbers, $p_1, p_2, p_3,\ldots,p_n$, then $p_1p_2p_3\cdots p_n+1$ is not divisible by any of the primes $p_i, i=1,2,3,\ldots,n.$ I dont understand why. Can somebody give me a hint or an Explanation ? Thanks.</p>
| szw1710 | 130,298 | <p>Simpler argument is that dividing by any $p_k$ we get the remainder $1$.</p>
|
2,877,080 | <p>Let A denote a commutative ring and let e denote an element of A such that $e^2 = e$.
How to prove that $eA \times (1 - e)A \simeq A$?
I thought that $\phi: A \mapsto eA \times (1 - e)A, \ \phi(a) = (ea, (1-e)a)$ is an isomorphism but I don't know how to prove that $\phi$ is a bijection.</p>
| paf | 333,517 | <p>It depends on the audience. I assume you talk to first year undergraduate math students. </p>
<p>Of course, you must include the precise statement of the theorems (in particular, insist on the condition $f(a)=f(b)$ in Rolle's theorem) and geometric interpretation (except maybe for Cauchy's MVT), maybe with their pr... |
264,770 | <p>If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then:</p>
<ol>
<li>May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = -3(1,2,3) \implies (1,2,3)$ or does the constant always go along with the vector?</li>
<li>If yes on question 1... | WhitAngl | 11,897 | <p>Starting with your question 2., the general relation $\|\lambda v\| = |\lambda| \|v\|$ always holds. From that you can answer positively to your first question (even if $\lambda$ is real) </p>
|
264,770 | <p>If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then:</p>
<ol>
<li>May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = -3(1,2,3) \implies (1,2,3)$ or does the constant always go along with the vector?</li>
<li>If yes on question 1... | Fly by Night | 38,495 | <p>You are perfectly entitled to "factorise" a vector, as you say $(-3,-6,-9) = -3(1,2,3).$ The important thing here is that this factorisation shows that the vectors $(-3,-6,-9)$ and $(1,2,3)$ are <a href="http://mathworld.wolfram.com/LinearlyDependentVectors.html" rel="nofollow">linearly dependent</a>. In the case of... |
2,332,277 | <p>First of all, note that $\frac{n^{n+1}}{(n+1)^n} \sim \frac{n}{e}$. </p>
<p><em>Question</em>: Is there $n>1$ such that $n^{n+1} \equiv 1 \mod (n+1)^n$?</p>
<p>There is an OEIS sequence for $n^{n+1}\mod (n+1)^n$: <a href="https://oeis.org/A176823" rel="nofollow noreferrer">https://oeis.org/A176823</a>. </p>
<... | kotomord | 382,886 | <p>Use the previous answer for the
$n = 4k + 1$</p>
<p>$n^{n+1}-1 = k \dot (n-1)$</p>
<p>For the $n=2*k$ </p>
<p>$n-1$ and $n+1$ is coprime => $n^{n+1}-1 = (n-1)(n+1)^n *k$. By the asymptotical equation from the comments k = 0.</p>
<p>For the $n=4k+3$. $\frac{n-1}{n}$ and $n+1$ is coprime $n^{n+1}-1 = k* \frac{... |
284,809 | <p>$F$ is a field and $F[X^2, X^3]$ is a subring of $F[X]$, the polynomial ring. I need to show that nonzero prime ideals of $F[X^2, X^3]$ are maximal.</p>
<p>A classmate suggested taking a nonzero prime ideal $\mathfrak{p}$ of $F[X^2, X^3]$ and embedding $F[X^2,X^3]/\mathfrak{p} \hookrightarrow F[X]/(\mathfrak{p})$ a... | Fabian | 7,266 | <p>For problems of this kind it is often convenient to treat the $\mathbf{x}$ and $\mathbf{x}^H$ as independent variables (in the end they are linearly related to real and imaginary part of $x$). So we want to minimize
$$E= \mathbf{x}^H Q \mathbf{x} - (\mathbf{x}^H \mathbf{b} + \mathbf{b}^H \mathbf{x}) +1.$$</p>
<p>I... |
430,629 | <p>The <strong>full linear monoid</strong> <span class="math-container">$M_N(k)$</span> of a field <span class="math-container">$k$</span> is the set of <span class="math-container">$N \times N$</span> matrices with entries in <span class="math-container">$k$</span>, made into a monoid with matrix multiplication.
A <st... | Benjamin Steinberg | 15,934 | <p>At least over an algebraically field, the polynomial representations are precisely the representations of <span class="math-container">$M_n(k)$</span> as an algebraic monoid and I don’t see immediately why that would change over a nonalgebraically closed field, but I am not an expert.</p>
<p>To classify irreducible ... |
1,513,373 | <p>Let M be a cardinal with the following properties:<br>
- M is regular<br>
- $\kappa < M \implies 2^\kappa < M$<br>
- $\kappa < M \implies s(\kappa) < M$ where $s(\kappa)$ is the smallest strongly inaccessible cardinal strictly greater than $\kappa$ </p>
<p>My question is: Is M a Mahlo cardinal ? If s... | Wojowu | 127,263 | <p>No, your condition doesn't imply Mahloness.</p>
<p>First, note that your first two conditions simply state that $M$ is inaccessible, and the third one gives that $M$ is limit of inaccessibles. Now consider the first cardinal $M_0$ which is inaccessible limit of inaccessibles. That is, every inaccessible below $M_0$... |
2,134,928 | <p>Let <span class="math-container">$ \ C[0,1] \ $</span> stands for the real vector space of continuous functions <span class="math-container">$ \ [0,1] \to [0,1] \ $</span> on the unit interval with the usual subspace topology from <span class="math-container">$\mathbb{R}$</span>. Let <span class="math-container">$$\... | Daniel Pietrobon | 17,824 | <p>Suppose she had 6 correct answers. That's $6 \times 3 - 4 = 18-4 = 14$. Nope too low!</p>
<p>What about 7? $7 \times 3 - 3 = 21-3 = 18$. Oh, we got it.</p>
|
1,849,577 | <p>I recently asked for <a href="https://math.stackexchange.com/questions/1848739/a-topology-on-the-set-of-lines">natural topologies on the set of lines</a> in $\mathbb R^2$. Now I'm aiming for a similar question on the set $S_p$ of conic sections in $\mathbb R^2$ sharing the same focus $p$ (but not necessary having th... | MvG | 35,416 | <p>It depends a lot on what exactly you want to call a conic. My first impulse was along the same lines as what N.H. wrote in a comment (and later in an answer): six numbers to describe a conic, but scalar multiples describe the same conic, so this looks like $\mathbb P^5$. But then N.H. went on to exclude degenerate c... |
211,803 | <p>I ended up with a differential equation that looks like this:
$$\frac{d^2y}{dx^2} + \frac 1 x \frac{dy}{dx} - \frac{ay}{x^2} + \left(b -\frac c x - e x \right )y = 0.$$
I tried with Mathematica. But could not get the sensible answer. May you help me out how to solve it or give me some references that I can go over... | Pragabhava | 19,532 | <p>Series solution around zero:</p>
<p>The point $x= 0$ is a regular singular. Taking the anzats
$$
y(x) = \sum_{n=0}^\infty q_n x^{n+s}
$$
we have
$$
\sum_{n=0}^\infty [(n+s)^2 - a]q_n x^{n+s-2} -\sum_{n=0}^\infty c q_n x^{n+s-1} + \sum_{n=0}^\infty b q_n x^{n+s} - \sum_{n=0}^\infty e q_n x^{n+s+1} =
$$
\begin{multli... |
1,460,488 | <p>There are $4$ girls and $3$ boys but there are only $5$ seats. How many ways can you seat the $3$ boys together?</p>
<p>The order of the seat matters, for example:
there's the order
$B_1$ $B_2$ $B_3$ $G_2$ $G_4$
and there's
$B_2$ $B_3$ $B_1$ $G_2$ $G_4$</p>
<p>Here's my answer:
There are $3!$ ways to seat the $3$ ... | Tejus | 274,219 | <p>First Take all of the boys in a group say $A$. $A$ contains $B_1$, $B_2$ and $B_3$. Then select two girls from $4 \Rightarrow$ $^4C_2$. Now we got $2$ girls and boys (in $A$) which makes it $5$. We can arrange $B_1$ $B_2$ $B_3$ inside $A$ in $3!$ ways . and we can arrange $A$ and the $2$ girls in $3!$ ways . Hence t... |
944,840 | <p>For vectors u, w, and v in a vector space V, I am trying to prove:</p>
<p>If $u + w = v + w$ then $u = v$</p>
<p><strong>without</strong> using the additive inverse and only using the 8 axioms which define a vector space. I am coming up short. I don't see how to do this without assuming that if $u + w = v + w$ the... | Berci | 41,488 | <p>You will just have to add a $w'$ indeed, namely, try $w':=(-1)\cdot w$ and use
$$(\alpha+\beta)w=\alpha w+\beta w$$
for certain $\alpha,\beta$ scalars.</p>
|
1,286,306 | <p>Suppose that $a_1,...,a_n,b_1,...,b_n ∈ F $ are such that $\sum a_ib_i = 1_F$. </p>
<p>Let $J : F^n → F^n $ be the linear transformation whose standard matrix has $ij^{th}$ entry $a_ib_j$. </p>
<p>Prove that $J^2 = J$.</p>
<p>So I think I've figured out that the index in the matrix $F^2$ given by </p>
<p>$u_{ij... | Lost in a Maze | 77,255 | <p>Some remarks for proceeding:</p>
<ul>
<li>To show $H$ is complete, note that $[0,1]^\infty = \{(x_1, x_2,\ldots)\,:\, x_i \in [0,1]\}$ is complete. Therefore, it suffices to show that $H$ is closed (a closed bounded subset of a complete space is also complete).</li>
<li>To show $H$ is totally bounded, given $\epsil... |
2,485,529 | <p>The integral is
$$\int{\left[\frac{\sin^8(x) - \cos^8(x)}{1 - 2 \sin^2(x)\cos^2(x)}\right]}dx$$</p>
| Humam | 470,705 | <p>this integer can be reduced using the double angle sine and cosine formula.</p>
<p>let
$$ I=\int \frac{sin^8(x)-cos^8(x)}{1-2sin^2(x)cos^2(x)}dx $$
$$ I=\int \frac{(sin^4(x)+cos^4(x))(sin^4(x)-cos^4(x))}{1-2sin^2(x)cos^2(x)}dx $$
$$ I=\int \frac{(sin^4(x)+2sin^2(x)cos^2(x)+cos^4(x)-2sin^2(x)cos^2(x))(sin^4(x)-cos^4... |
137,501 | <p>P. P. Palfy proved that a primitive solvable subgroup of $S_n$ has order bounded by $24^{-1/3} n^{3.24399\dots}$ (in: Pálfy, P. P.
A polynomial bound for the orders of primitive solvable groups.
J. Algebra 77 (1982), no. 1, 127–137. )</p>
<p>This is sharp (and notice that for $n$ prime, the affine group of the lin... | Nick Gill | 801 | <p>This answer fleshes out observations made in comments above. The result below is an analogue of Palfy's theorem for nilpotent primitive groups, as requested.</p>
<blockquote>
<p><strong>Prop</strong>. Let $N<S_n$ be a nilpotent primitive group. Then $n$ is prime and $N$ is cyclic of order $n$.</p>
</blockquote... |
186,638 | <p>$f(x)=\max(2x+1,3-4x)$, where $x \in \mathbb{R}$. what is the minimum possible value of $f(x)$.</p>
<p>when, $2x+1=3-4x$, we have $x=\frac{1}{3}$</p>
| copper.hat | 27,978 | <p>The function $f$ is convex and its subdifferential is given by
$$\partial f(x) =
\begin{cases}
\{-4\}, & x<\frac{1}{3}, \\ \, [-4,2], & x = \frac{1}{3}, \\ \{2\}, & x>\frac{1}{3}.
\end{cases}$$Since $f$ is convex, then $\hat{x}$ minimizes $f$ iff $0 \in \partial f (\hat{x})$. It follows that the mi... |
3,863,495 | <p>This is my solution to an old exam problem that I'd appreciate some feedback on. The problem:</p>
<blockquote>
<p>Let <span class="math-container">$f:[0,\infty)\to\mathbb{R}$</span>, <span class="math-container">$f\geq 0$</span> and <span class="math-container">$\int _0^{\infty} f(x) dx=L<\infty;$</span> that is,... | copper.hat | 27,978 | <p>Pick <span class="math-container">$\epsilon>0$</span> and choose <span class="math-container">$T$</span> such that <span class="math-container">$\int_T^\infty f(x)dx < {1 \over 2} \epsilon$</span>. Suppose <span class="math-container">$R >T$</span>.</p>
<p><span class="math-container">\begin{eqnarray}
\int_... |
80,456 | <p>Given an array <code>sel</code> and an index position <code>i0</code>, how can I find the position of the nearest (left or right) nonzero element?
I'm able to do it with a loop and a couple of awful If's, but I was looking for a functional way...</p>
<pre><code> lr=Length[sel];
For[i = 0, i <= lr, i++,
... | Jinxed | 24,763 | <p>Try this one:</p>
<pre><code>nearestNonNull[lst_, i_] :=
First@MinimalBy[
Select[MapIndexed[Flatten@{#1 != 0, #2} &, lst],
TrueQ@First@# &][[All, 2]], Abs[i - #] &]
sel = RandomInteger[{0, 10}, 10^4];
nearestNonNull[sel, 1234];
</code></pre>
|
80,456 | <p>Given an array <code>sel</code> and an index position <code>i0</code>, how can I find the position of the nearest (left or right) nonzero element?
I'm able to do it with a loop and a couple of awful If's, but I was looking for a functional way...</p>
<pre><code> lr=Length[sel];
For[i = 0, i <= lr, i++,
... | LLlAMnYP | 26,956 | <p>Here's a generalization of the above to work with arrays (lists) of arbitrary depth. Also avoids checking the element at your specified position (something which may or may not be desired).</p>
<pre><code>nearestNZP =
Function[{array, i},
MinimalBy[
Flatten[MapIndexed[
If[#1 != 0 && #2 != i... |
4,428,142 | <p>Applying integration by parts splits the integral into 3 integrals,
<span class="math-container">$\displaystyle \begin{aligned}I&=\int_{0}^{1} \frac{\sin ^{-1} x \ln (1+x)}{x^{2}} d x\\&=-\int_{0}^{1} \sin ^{-1} x \ln (1+x) d\left(\frac{1}{x}\right) \\&=-\left[\frac{\sin ^{-1} x \ln (1+x)}{x}\right]_{0}^... | Quanto | 686,284 | <p>A self-contained solution
<span class="math-container">\begin{align}
I=&\int_{0}^{1} \frac{\sin ^{-1} x \ln (1+x)}{x^{2}} d x
=\int_{0}^{1} \sin ^{-1}x \>d\left( \ln x -\frac{1+x}x \ln (1+x)\right)\\
\overset{ibp} =&\> -\pi \ln 2-{\int_{0}^{1} \frac{\ln \frac x{1+x}-\frac1x \ln(1+x)}{\sqrt{1-x^{2}}}}\... |
3,074,900 | <h2>Problem</h2>
<p>When proving one result in the statistical learning theory course, the instructor uses
<span class="math-container">$$
\mathbb{E}[\mathbb{E}[X\vert Y,Z]\vert Z]=\mathbb{E}[X\vert Z]
$$</span>
but I am not sure why this is true.</p>
<h2>What I Have Done</h2>
<p>I know I could do the following
<spa... | Masoud | 653,056 | <p>by tower property </p>
<p>if <span class="math-container">$F_1 \subset F_2$</span> so <span class="math-container">$E(E(X|F_2)|F_1)=E(X|F_1)$</span></p>
<p>now </p>
<p><span class="math-container">$\mathbb{E}[\mathbb{E}[X\vert Y,Z]\vert Z]=\mathbb{E}[\mathbb{E}[X\vert \sigma (Y,Z)]\vert \sigma(Z)]$</span></p>
<... |
3,574,460 | <p>Suppose <span class="math-container">${X_0, X_1, . . . , }$</span> forms a Markov chain with state space S. For any n ≥ 1
and <span class="math-container">$i_0, i_1, . . . , ∈ S$</span>, which conditional probability, <span class="math-container">$P(X_0 = i_0|X_1 = i_1)$</span> or <span class="math-container">$P(X_... | oso_hormiguero | 1,053,142 | <p><span class="math-container">\begin{aligned}
P(X_0 = i_0|X_1 = i_1, . . . , X_n = i_n) &= \frac{P(X_0 = i_0, X_1 = i_1, . . . , X_n = i_n)}{P(X_1 = i_1, . . . , X_n = i_n)} & \text{(conditional probability)}\\
&= \frac{P(X_0 = i_0)P( X_1 = i_1 |X_0 = i_0)\cdots P(X_n = i_n | X_{n-1} = i_{n-1})}{P(X_1 = i... |
1,035,877 | <p>I'd really appreciate if someone could help me so I could get going on these problems, but this is confusing me... and it's been holding me up for the last couple hours. </p>
<p>How can I find the volume of the solid when revolving the region bounded by $y=1-\frac{1}{2}x$, $y=0$, and $x=0$ about the line $ x=-1$? H... | Mr. Math | 70,964 | <p>Hint</p>
<p>$$volume=\int\pi y^2dx$$</p>
<p>use limites $x=-1$ and $x=0$</p>
|
2,609,252 | <p>like the title said i'm looking for the best way for me(a 15 year old) to go about learning calculus, thank you :)</p>
| Ski Mask | 503,445 | <p>Go straight to Khan Academy and start with their beginner courses. It's by far one of the most effective ways to learn things. I'm a CS student in university and keep coming back to Khan Academy to help with things from limits or L'Hopital's Rule. So for someone who wants to start learning Calculus Khan Academy is t... |
2,609,252 | <p>like the title said i'm looking for the best way for me(a 15 year old) to go about learning calculus, thank you :)</p>
| Wessel de Zeeuw | 160,433 | <p>You could take a look at the online Pre-Calculus course of the Technical University in Delft. <a href="https://online-learning.tudelft.nl/courses/pre-university-calculus/" rel="nofollow noreferrer">https://online-learning.tudelft.nl/courses/pre-university-calculus/</a>
I think this will fit your academic and enthusi... |
322,134 | <p>$$2e^{-x}+e^{5x}$$</p>
<p>Here is what I have tried: $$2e^{-x}+e^{5x}$$
$$\frac{2}{e^x}+e^{5x}$$
$$\left(\frac{2}{e^x}\right)'+(e^{5x})'$$</p>
<p>$$\left(\frac{2}{e^x}\right)' = \frac{-2e^x}{e^{2x}}$$
$$(e^{5x})'=5xe^{5x}$$</p>
<p>So the answer I got was $$\frac{-2e^x}{e^{2x}}+5xe^{5x}$$</p>
<p>I checked my answ... | lsp | 64,509 | <p>derivative of $2e^{-x} = -2e^{-x}$</p>
<p>derivative of $e^{5x} = 5e^{5x}$</p>
<p><strong>In general the derivative of $e^{ax} = ae^{ax}$, where $a$ is a constant.</strong></p>
|
3,218,525 | <p>Let <span class="math-container">$f:[0,1] \to [0, \infty)$</span> is a non-negative continuous function so that <span class="math-container">$f(0)=0$</span> and for all <span class="math-container">$x \in [0,1]$</span> we have <span class="math-container">$$f(x) \leq \int_{0}^{x} f(y)^2 dy$$</span><br>
Now consider ... | Henk | 334,507 | <p>A bit of a different approach I believe also to be correct: since <span class="math-container">$f(x)\leq\int_0^xf(y)^2dy$</span> for all <span class="math-container">$x\in[0,1]$</span>, we can say that <span class="math-container">$f$</span> is bounded from above (i.e. <span class="math-container">$f(x)\leq\bar{f}(x... |
142,993 | <p>I'm challenging myself to figure out the mathematical expression of the number of possible combinations for certain parameters, and frankly I have no idea how.</p>
<p>The rules are these:</p>
<p>Take numbers 1...n. Given m places, and with <em>no repeated digits</em>, how many combinations of those numbers can be ... | Christian Blatter | 1,303 | <p>Assume that $(0,0)$ is a white square. Call a white square <em>even</em> if it has even coordinates and <em>odd</em>, if it has odd coordinates; there are $4^2=16$ of each. The first square you can choose in $32$ ways; assume you pick an even one. The second square either is even, and there are $3^2=9$ left of these... |
160,801 | <p>Here is a vector </p>
<p>$$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$</p>
<p>Here is a matrix</p>
<p>$$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$</p>
<p>Is there a simple way to determine whether the vector is an eigenvector of this matrix?</p>
<p>Here is some code for your conven... | Carl Woll | 45,431 | <p>You could use <a href="http://reference.wolfram.com/language/ref/MatrixRank" rel="noreferrer"><code>MatrixRank</code></a>. Here is a function that does this:</p>
<pre><code>eigenvectorQ[matrix_, vector_] := MatrixRank[{matrix . vector, vector}] == 1
</code></pre>
<p>For your example:</p>
<pre><code>eigenvectorQ[h... |
3,014,085 | <p>I am trying to isolate y in this equation:
<span class="math-container">$$-4/3·\ln(|y-60|)=x+c$$</span></p>
<p>If I use a cas-tool to isolate <span class="math-container">$y$</span>, I get:</p>
<p><span class="math-container">$$60.-(2.71828182846)^{−0.75*x-0.75*c}=y$$</span></p>
<p>If I try isolating <span class="m... | user | 505,767 | <p><strong>HINT</strong></p>
<p>A trivial case is when <span class="math-container">$z$</span> is real.</p>
<p>Excluding the trivial case, let consider different cases, for example for <span class="math-container">$z$</span> in the first quadrant we need that <span class="math-container">$z-2$</span> is in the second... |
353,480 | <p>Is $f(x)=\ln(x)$ uniformly continuous on $(1,+\infty)$? If so, how to show it?</p>
<p>I know how to show that it is not uniformly continuous on $(0,1)$, by taking $x=\frac{1}{\exp(n)}$ and $y = \frac{1}{\exp(n+1)}$.</p>
<p>Also, on which interval does $\ln(x)$ satisfy the Lipschitz condition?</p>
| Community | -1 | <p>Let $x,y\in(1,+\infty)$ then by mean value theorem there's $z$ such that
$$\log(x)-\log(y)=\frac{1}{z}(x-y)$$
hence we have
\begin{array}\\
|\log(x)-\log(y) |&\leq\sup_{z>1}\frac{1}{z}|x-y|\\&\leq|x-y|
\end{array}
so the function $\log$ is uniformly continuous on the interval $(1,+\infty)$ since it sati... |
2,354,036 | <p>$\require{AMScd}\def\colim{\text{colim}}$I need this result in less generality, but I'd be happy to know this stronger version holds.</p>
<p>Let $\{\alpha_c : Fc \to Gc\}$ be arrows in a category $D$, indexed by the objects of a category $C$, for two functors $F,G: C\to D$. </p>
<p>Let $A\subseteq C$ be a dense su... | Eric Wofsey | 86,856 | <p>This is very very false. You can simply take any natural transformation $Fi\to Gi$, and then define $\alpha_c$ for each $c\not\in A$ to be any map $Fc\to Gc$ at all. It would be an enormous coincidence if the latter satisfied naturality.</p>
<p>For a simple example, let $C=D=Set$, $F=G=1_{Set}$, and let $A$ consi... |
1,556,747 | <p>$$\text{a)} \ \ \sum_{k=0}^{\infty} \frac{5^{k+1}+(-3)^k}{7^{k+2}}\qquad\qquad\qquad\text{b)} \ \ \sum_{k=1}^{\infty}\log\bigg(\frac{k(k+2)}{(k+1)^2}\bigg)$$</p>
<p>I am trying to determine the convergence values. I tried with partial sums and got stuck...so I am thinking the comparison test...Help</p>
| Mark Viola | 218,419 | <p>HINTS:</p>
<p>$$\frac{5^{k+1}+(-3)^{k}}{7^{k+2}}=\frac5{49}\left(\frac57\right)^k+\frac1{49}\left(-\frac{3}{7}\right)^k$$</p>
<p>$$\log\left(\frac{k(k+2)}{(k+1)^2}\right)=(\log (k)-\log (k+1))+(\log (k+2)-\log (k+1))$$</p>
<p>Then, telescope the two series.</p>
|
438,166 | <p>Let $X$ be a bounded connected open subset of the $n$-dimensional real Euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support in $X$. </p>
<p>Does the closure of this operator generate a strongly continuous semigroup on $C_0(X)$ endowed with t... | DeM | 113,757 | <p>I completely agree with the previous answer but I would like to add that in general there are even <em>infinitely many</em> extensions of the Laplacian - not only Dirichlet and Neumann. E.g., the Laplacian with all Robin-type boundary conditions
$$
\frac{\partial u}{\partial n}=pu_{|\partial X}
$$
will do the job.</... |
1,129,712 | <p>So I'm complete stuck with something. I know it the following statements are true (or at least the seem to be from the results that I got from messing around with it a bit on MATLAB), but I don't understand why they are true or how to show so.
Let $A$ be and $m$X$n$ matrix. Show that:</p>
<p>a) if $x \in N(A^TA)$ t... | egreg | 62,967 | <p>a) By definition $Ax\in R(A)$; on the other hand, $A^TAx=0$, by assumption, so $Ax\in N(A^T)$.</p>
<p>b) It is clear that $N(A)\subseteq N(A^TA)$. Suppose $x\in N(A^TA)$; then $x^TA^TAx=0$ as well, so $(Ax)^T(Ax)=0$, which implies $Ax=0$.</p>
<p>c) The rank-nullity theorem says that, if $B$ is an $m\times n$ matri... |
2,120,510 | <blockquote>
<p>Out of <span class="math-container">$180$</span> students, <span class="math-container">$72$</span> have Windows, <span class="math-container">$54$</span> have Linux, <span class="math-container">$36$</span> have both Windows and Linux and the rest (<span class="math-container">$18$</span>) have OS X. W... | Kanwaljit Singh | 401,635 | <p>Hint -</p>
<p>Case 1 -</p>
<p>Sum of probabilities that no one has windows, exactly one and exactly two students don't have windows.</p>
<p>Case 2 -</p>
<p>At least one have OS X = 1 - No one have OS X</p>
<p>$= 1 - \frac{\binom{162}{15}}{\binom{180}{15}}$</p>
|
2,120,510 | <blockquote>
<p>Out of <span class="math-container">$180$</span> students, <span class="math-container">$72$</span> have Windows, <span class="math-container">$54$</span> have Linux, <span class="math-container">$36$</span> have both Windows and Linux and the rest (<span class="math-container">$18$</span>) have OS X. W... | Dove | 402,662 | <ol>
<li><p>For your first question, it will be the sum of probabilities that no one has windows, exactly one student doesn't have windows, and exactly two students don't have windows.</p></li>
<li><p>P=1- P(no one has X)</p></li>
</ol>
|
2,604,093 | <p>I would like to study the convergence of the series:</p>
<p>$$\sum_{n=1}^\infty \frac{\log n}{n^2}$$</p>
<p>I could compare the generic element $\frac{\log n}{n^2}$ with $\frac{1}{n^2}$ and say that
$$\frac{1}{n^2}<\frac{\log n}{n^2}$$ and $\frac{1}{n^2}$ converges but nothing more about.</p>
| José Carlos Santos | 446,262 | <p>By <a href="https://en.wikipedia.org/wiki/Cauchy_condensation_test" rel="nofollow noreferrer">Cauchy's condensation test</a>, your series converges if and only if$$\sum_{n=1}^\infty\frac{2^n\log2^n}{2^{2n}}$$converges. But this series is equal to$$\sum_{n=1}^\infty\frac{n\log2}{2^n}$$which clearly converges.</p>
|
2,604,093 | <p>I would like to study the convergence of the series:</p>
<p>$$\sum_{n=1}^\infty \frac{\log n}{n^2}$$</p>
<p>I could compare the generic element $\frac{\log n}{n^2}$ with $\frac{1}{n^2}$ and say that
$$\frac{1}{n^2}<\frac{\log n}{n^2}$$ and $\frac{1}{n^2}$ converges but nothing more about.</p>
| zhw. | 228,045 | <p>Repeat after me: $\ln n\to \infty$ more slowly than any positive power of $n.$ In other words,</p>
<p>$$\frac{\ln n}{n^p} \to 0\,\text { for any } p> 0.$$</p>
<p>Once you have absorbed this, you'll know such things as</p>
<p>$$\frac{\ln n}{n^2}< \frac{n^{1/2}}{n^2} = \frac{1}{n^{3/2}}$$</p>
<p>for large $... |
4,537,050 | <p>Question 2 of Chapter 14 in Spivak's <em>Calculus</em> reads as follows:</p>
<blockquote>
<p>For each of the following <span class="math-container">$f$</span>, if <span class="math-container">$F(x)=\int_0^xf$</span>, at which points <span class="math-container">$x$</span> is <span class="math-container">$F'(x)=f(x)$... | Matt Werenski | 733,040 | <p>Having not read the book, I can't speak to what the "rules" are for integration, but it probably falls into one of the two camps below.</p>
<p><strong>Riemann integral rules</strong> If the book is using Riemann integration then we can construct a series of step functions <span class="math-container">$(\ph... |
886,243 | <p>Evaluate</p>
<p><img src="https://latex.codecogs.com/gif.latex?%0A%24%245050%20%5Cfrac%20%7B%5Cleft(%20%5Csum%20_%7Br%3D0%7D%5E%7B100%7D%20%5Cfrac%20%7B%7B100%5Cchoose%20r%7D%7D%7B50r%2B1%7D%5Ccdot%20(-1)%5Er%5Cright)%20-%201%7D%7B%5Cleft(%20%5Csum%20_%7Br%3D0%7D%5E%7B101%7D%5Cfrac%7B%7B101%5Cchoose%20r%7D%7D%7B50r... | RE60K | 67,609 | <p><strong>Use of sequence and series is not suggested; and a possible way is outlined, which is very long and useless, this is due to the fact of bounded multiplication of r with 50 in denominator, if the r would have been free, it would be easy to use this method to caluculate:</strong></p>
<h2>Brute Force</h2>
<p>... |
3,968,905 | <p>I am trying to prove this:</p>
<p><span class="math-container">$\bullet$</span> Prove that <span class="math-container">$\Delta(\varrho_\epsilon \star u) = \varrho_\epsilon \star f $</span> in the sense of distributions, if <span class="math-container">$\Delta u = f$</span> in the sense of distributions, <span class... | RicardoMM | 730,135 | <p>Using wolfram alpha, all the solutions are given by the expression : <span class="math-container">$$\begin{cases}y=\frac{x-1}{x+1}, x\neq -1 \\y = \frac {x+3}{x+1}, x\neq -1 \end{cases}$$</span>
For example, the solutions for <span class="math-container">$x=1$</span> are <span class="math-container">$y=0$</span> and... |
3,968,905 | <p>I am trying to prove this:</p>
<p><span class="math-container">$\bullet$</span> Prove that <span class="math-container">$\Delta(\varrho_\epsilon \star u) = \varrho_\epsilon \star f $</span> in the sense of distributions, if <span class="math-container">$\Delta u = f$</span> in the sense of distributions, <span class... | tiredsoldat | 868,820 | <p>The problem is that you are trying to solve this by simplifying instead of substituting for x and y. Hint to this problem is to first look at the graph of it on desmos perhaps. From this you can see the only integer solutions will ever be found on the axis'. Hence where x=0 or where y=0. This is the easiest method t... |
2,120,763 | <p>I've been given this problem:</p>
<p>Prove that a subordinate matrix norm is a matrix norm, i.e. </p>
<p>if $\left \|. \right \|$ is a vector norm on $\mathbb{R}^{n}$, then $\left \| A \right \|=\max_{\left \| x \right \|=1}\left \| Ax \right \|$ is a matrix norm</p>
<p>I don't even understand the question, and ... | lab bhattacharjee | 33,337 | <p>Let the highest power of prime $p$ that divides $a,b,c$ be $A,B,C$ respectively.</p>
<p>So, the highest power of prime $p$ that divides the GCD will be min$(A,B,C)$</p>
<p>and the highest power of prime $p$ that divides the LCM will be max$(A,B,C)$</p>
<p>We need min$(A,B,C)+$max$(A,B,C)=A+B+C$ for any prime th... |
1,624 | <p>For example, to change the color of each pixel to the mean color of the three channels, I tried</p>
<pre><code>i = ExampleData[{"TestImage", "Lena"}];
Mean[i]
</code></pre>
<p>but it just remains unevaluated:</p>
<p><img src="https://i.stack.imgur.com/K1RRR.png" alt="enter image description here"></p>
<p>How can... | cormullion | 61 | <p>The ImageApply function applies any suitable Mathematica function to every pixel in an image. You just have to specify the transformation you want to make. I recently asked this: <a href="https://mathematica.stackexchange.com/questions/207/image-levels-how-to-alter-exposure-of-dark-and-light-areas">Image levels: how... |
1,624 | <p>For example, to change the color of each pixel to the mean color of the three channels, I tried</p>
<pre><code>i = ExampleData[{"TestImage", "Lena"}];
Mean[i]
</code></pre>
<p>but it just remains unevaluated:</p>
<p><img src="https://i.stack.imgur.com/K1RRR.png" alt="enter image description here"></p>
<p>How can... | Brett Champion | 69 | <p>There are also several built-in effects available through <code>ImageEffect</code>. For example:</p>
<pre><code>ImageEffect[ExampleData[{"Image","Lena"}], #]& /@
{"Charcoal", {"OilPainting", 10}, {"Posterization", 5}}
</code></pre>
<blockquote>
<p><img src="https://i.stack.imgur.com/y18SQ.png" alt="Mat... |
244,492 | <p>Find $m \in \mathbb R$ for which the equation $|x-1|+|x+1|=mx+1$ has only one unique solution. When does a absolute value equation have only 1 solution?</p>
<p>I solved for $x$ in all 4 cases and got $x=\frac{1}{-m-2},x=\frac{1}{2-m},x=\frac{1}{m},x=-\frac{3}{m}$</p>
| Clive Newstead | 19,542 | <p><strong>Method 1:</strong> Draw a graph to find the answer, and then prove that your answer holds.</p>
<p><strong>Method 2:</strong> Solve the equation in the separate cases</p>
<ul>
<li>$x \le -1$ (so that $|x+1|=-(x+1)$ and $|x-1|=-(x-1)$)</li>
<li>$-1 \le x \le 1$ (so that etc.)</li>
<li>$x \ge 1$ (etc.)</li>
<... |
90,263 | <p>Let $\mathcal{E} = \lbrace v^1 ,v^2, \dotsm, v^m \rbrace$ be the set of right
eigenvectors of $P$ and let $\mathcal{E^*} = \lbrace \omega^1 ,\omega^2,
\dotsm, \omega^m \rbrace$ be the set of left eigenvectors of $P.$ Given any two
vectors $v \in \mathcal{E}$ and $ \omega \in \mathcal{E^*}$ which correspond to
t... | Neal | 20,569 | <p>Culture affects the student's learning style, classroom expectations, attitude toward learning, and is even correlated with their mathematical background. Teaching a classroom full of poor, mostly black or hispanic kids who are first-generation college students is very different from teaching a classroom full of up... |
1,605,281 | <p><a href="https://i.stack.imgur.com/43uoh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/43uoh.png" alt="The question about finding the exact value of the sine of the angle between (PQ) and the plane"></a></p>
<p>I have done part (a). For part (b), I know the principle of how to do it, I tried to... | Brian M. Scott | 12,042 | <p>You have the essential idea, but you’ve stated it rather badly. To show that $f$ is not continuous, you need only find an open set $U$ such that $f^{-1}[U]$ is not open. You are correct in thinking that taking $U=[a,b)$ will work, but you need to say that that’s what you’re doing:</p>
<blockquote>
<p>Let $U=[a,b)... |
3,856,370 | <p>this is the result in the book(Discrete mathematics and its applications) I was reading.</p>
<ol>
<li><span class="math-container">$n^d\in O(b^n)$</span></li>
</ol>
<p>where <span class="math-container">$b>1$</span> and <span class="math-container">$d$</span> is positive</p>
<p>and</p>
<ol start="2">
<li><span cl... | Claude Leibovici | 82,404 | <p>If you like special functions
<span class="math-container">$$I=\int\left(\frac{\sin x}{x^3}-\frac1{x^2}\right)\,dx=-\frac{\text{Si}(x)}{2}-\frac{\sin (x)}{2 x^2}+\frac{1}{x}-\frac{\cos (x)}{2 x}$$</span>
<span class="math-container">$$J(p)=\int_0^p\left(\frac{\sin x}{x^3}-\frac1{x^2}\right)\,dx=-\frac{\text{Si}(p)}{... |
129,295 | <p>$$\int{\sqrt{x^2 - 2x}}$$</p>
<p>I think I should be doing trig substitution, but which? I completed the square giving </p>
<p>$$\int{\sqrt{(x-1)^2 -1}}$$</p>
<p>But the closest I found is for</p>
<p>$$\frac{1}{\sqrt{a^2 - (x+b)^2}}$$ </p>
<p>So I must add a $-$, but how? </p>
| Kns | 27,579 | <p>I dont know i am right or wrong but i can do this example without using trigonometric substitution in following way,
\begin{align*}
\int\sqrt{x^{2}-2x}dx &=\int\sqrt{x^{2}-2x+1-1}dx\
&=\int\sqrt{(x-1)^{2}-1^{2}}dx\
&=\frac{x}{2}\sqrt{(x-1)^{2}-1}-\frac{1}{2}\log |x... |
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