All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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Prove: $\int_{0}^{\infty} \sin (x^2) dx$ converges.

$\sin(x^2)$ is an example for a function which its limit when $x \to \infty$ is not $0$, and still its integral from $0$ to $\infty$ is finite. I'd like your help with understanding why and a ...
14
votes
5answers
851 views

Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$

I'd like your help with proving that $$\int_0^1 \frac{\ln x }{x-1}d x=\sum_{n=1}^\infty \frac{1}{n^2}.$$ I tried to use Fourier series, or to use a power series and integrate it twice but it didn't ...
14
votes
3answers
502 views

Prove that if $\int f^2$ and $\int( f'')^2$ converge, so does $\int (f')^2$

Question: Let $f: [a,\infty) \to \Bbb R \in C^2$ and the two following integrals converge: $$\int _a^\infty (f''(x))^2\,dx ,~~~~~~~~~ \int _a^\infty (f(x))^2\,dx$$ Prove that $\int ...
14
votes
2answers
367 views

Show $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

I have run a FORTRAN code and I have obtained strong evidence that $$\int_0^{\pi/3} \!\! \big((\sqrt{3}\cos\vartheta-\sin\vartheta)\sin\vartheta\big)^{\!1/2}\!\cos\vartheta \,d\vartheta ...
14
votes
6answers
473 views

Integrate: $ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $ without using complex analysis methods

Can this integral be solved without using any complex analysis methods: $$ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $$ Thanks.
14
votes
3answers
366 views

How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$

If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x)dx$ exists. How can I prove that $$\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left( x - ...
14
votes
4answers
434 views

Can we find a closed form for $ \int_{- \infty}^{\infty} \frac{\exp\left(-(a+bx)^2\right)}{1+\exp(x)}\mathrm dx$?

Can we find a closed form for this definite integral: $$ \int_{- \infty}^{\infty} \frac{\exp\left(-(a+bx)^2\right)}{1+\exp(x)}\mathrm dx $$
14
votes
2answers
2k views

A hard definite integral with trignometric

How could we get a closed form for this one? $$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x \right)\text{d}x}$$
14
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5answers
2k views

Computing the integral of $\log(\sin x)$

How to compute the following integral $$\int\log(\sin x)~dx~?$$
14
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4answers
295 views

Evaluate ${\int_0^\infty e^{-t} \log(\cos^2 t)}\,\mathrm dt$

After reading a textbook about integrals, my curiosity sparks about the following integral: $$\int_0^\infty e^{-t} \log(\cos^2 t)\, \mathrm dt$$ How to evaluate a closed form of this integral ? My ...
14
votes
3answers
1k views

$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

Trying to solve $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges. I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$ Would ...
14
votes
3answers
382 views

integral with $\log\left(\frac{x+1}{x-1}\right)$

I encountered a tough integral and I am wondering if anyone has any ideas on how to evaluate it. $$\displaystyle ...
14
votes
4answers
660 views

Interesting integral related to the Omega Constant/Lambert W Function

I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof. I looked through the site. If it is here and I overlooked it, I am sorry. ...
14
votes
3answers
563 views

Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ ...
14
votes
3answers
511 views

Show that a definite integral vanishes for all values of $a,b,c > 0$.

Show $$\int_0^\infty \int_0^\infty \frac{(ax-by) {\rm e}^{-x} {\rm e}^{-y}}{(a^2 x + b^2 y + c x y)^{\frac{3}{2}}} \,dx \,dy = 0$$ for any $a,b,c > 0$. I came upon the above double integral when ...
14
votes
2answers
148 views

Integral $\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz$ involving Dawson's integrals

I need you help with evaluating this integral: $$I=\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz,\tag1$$ where $F(x)$ represents Dawson's integral: $$F(x)=e^{-x^2}\int_0^x ...
14
votes
4answers
230 views

Compute $\int_{0}^{1}\left[\frac{2}{x}\right]-2\left[\frac{1}{x}\right]dx$

The question is to find $$\int_{0}^{1}\left[\dfrac{2}{x}\right]-2\left[\dfrac{1}{x}\right]dx,$$ where $[x]$ is the largest integer no greater than $x$, such as $[2.1]=2, \;[2.7]=2,\; [-0,1]=-1.$ Is ...
14
votes
4answers
1k views

Prove $\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \, dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \, dx$

Prove that: $(1)$$$\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \ dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \ dx$$ $(2)$$$\int_0^{\infty } \frac{1}{\sqrt{8 x^3+x+7}} \ dx>1$$ What I do for ...
14
votes
2answers
315 views

Integral involving $\coth (x)$: Maple and Mathematica disagree

I was posed an interesting integral. $$ \int_{-\infty}^{\infty}\left(\frac{\coth(x)}{x^{3}}-\frac{1}{3x^{2}}-\frac{1}{x^{4}}\right)dx .$$ The integral evaluates to ...
14
votes
2answers
342 views

$\lim_{a \to 0^{+}} \int_{0}^{a} \frac{1}{\sqrt{\cos(x)-\cos(a)}} \;dx=\frac{\pi}{\sqrt{2}}$

How can I prove that $$\lim_{a \to 0^{+}} \int_{0}^{a} \frac{1}{\sqrt{\cos(x)-\cos(a)}} \;dx=\frac{\pi}{\sqrt{2}}$$
14
votes
2answers
321 views

How to evaluate $\displaystyle\int_0^1\frac{\log^2(1+x)}x\mathrm dx$?

The definite integral $$\int_0^1\frac{\log^2(1+x)}x\mathrm dx=\frac{\zeta(3)}4$$ arose in my answer to this question. I couldn't find it treated anywhere online. I eventually found two ways to ...
14
votes
1answer
261 views

Proving a formula for $\int_0^\infty \frac{\log(1+x^{4n})}{1+x^2}dx $ if $n=1,2,3,\cdots$

I came across the formula $$\int_0^\infty \frac{\log \left(1+x^{4n} \right)}{1+x^2}dx = \pi \log \left\{2^n \prod_{k=1 ,\ k \text{ odd}}^{2n-1} \left(1+\sin \left( \frac{\pi k}{4n}\right) ...
14
votes
2answers
553 views

A Challenging Integral $\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx$

I encountered a strange integral with a strange result. $$\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx = \pi \log \left(\log (2) \right)$$ Believe it or not, the result agrees ...
14
votes
3answers
211 views

Sufficient bound to conclude limit has certain value. $\lim {\left( {\int_0^1 {{{dx} \over {1 + {x^n}}}} } \right)^n}=\frac 1 2 $

I am trying to show that $$\lim {\left( {\int\limits_0^1 {{{dx} \over {1 + {x^n}}}} } \right)^n}=\frac 1 2 $$ Now, this can be done as follows. Using $x\mapsto x^{-1}$ we get that $$\int\limits_0^1 ...
14
votes
1answer
358 views

Functions for which $\int f(g(x))\, \mathrm dx = f\left(\int g(x) \, dx\right)$

I was playing around with some integrals, and this question popped into my head: What functions exist such that the following is true? $$\int f(g(x))\;\mathrm dx = f\left(\int g(x)\;\mathrm ...
14
votes
1answer
294 views

Further our knowledge of a certain class of integral involving logarithms.

$\newcommand{\limitp}{\alpha}\newcommand{\innerp}{\beta}$I am fascinated by definite integrals. Exploring math.stackexchange, I have found many interesting integrals of the form $$ ...
14
votes
2answers
204 views

Evaluate $\sum_{n=1}^\infty 1/n^2$ using $\int_0^1 \int_0^1 \frac{\mathrm{d}x \, \mathrm{d}y}{1-xy}$

This paper http://math.ucsb.edu/~cmart07/Evaluating%20Integrals.pdf hints at a way to compute the sum $$ \sum_{n=1}^\infty \frac{1}{n^2} $$ by expanding it into the double integral $$\int_0^1 \int_0^1 ...
14
votes
1answer
197 views

Evaluation of definite Integral

Evaluate $$ \int_{\ln(0.5)}^{\ln(2)}\left( \frac{\displaystyle\sin x \frac{\sqrt{\sin^2(\cos x)+\pi e^{(x^4)}}}{1+(xe^{\cos x}\sin x)^2}+ 2\sin(x^2+2)\arctan\left(\frac{x^3}{3}\right) } ...
13
votes
5answers
844 views

A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$

What is the proof of the following: $$\int_{0}^{1} \left(\frac{\ln t}{1-t}\right)^2 \,\mathrm{d}t=\frac{\pi^2}{3} \>?$$
13
votes
4answers
1k views

Simpler way to compute a definite integral without resorting to partial fractions?

I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$ Is there a simpler way to do this ?
13
votes
5answers
747 views

A simple way to evaluate $\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx$?

I am currently trying to show that $\int_{-\infty}^\infty \cos(x^2) \, \mathrm dx = \sqrt{\frac{\pi}{2}}$ and the last integral I have to evaluate is $$\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx.$$ ...
13
votes
2answers
402 views

Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$

Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
13
votes
4answers
901 views

Finding $\int_0^{\pi/2} \sin x\,dx$

I'm interested in why $$\int_0^{\pi/2} \sin x\,dx = 1.$$ I know how to do the integral the conventional way but am more interested in what makes radians special for this problem. If we instead compute ...
13
votes
6answers
561 views

How closely can we estimate $\sum_{i=0}^n \sqrt{i}$

By looking at an integral and bounding the error?
13
votes
3answers
778 views

How do I evaluate this integral $\int\limits_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x$?

Show that $$\int\limits_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x =\frac{{{\pi^3}}}{{15}}+2\pi{\ln^2}\left({\frac{{1+\sqrt 5 }}{2}}\right).$$ I don't have any idea how to start, ...
13
votes
7answers
611 views

Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. [duplicate]

Can anyone give me a hint to evaluate this integral? $$\int_0^\infty \frac{dx}{1+x^4}$$ I know it will involve the gamma function, but how?
13
votes
4answers
623 views

$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \mathrm {d}x$ Evaluate Integral

Here is a fun integral I am trying to evaluate: $$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$ I thought about integrating by parts $2n$ times and then using ...
13
votes
4answers
440 views

How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?

How do I integrate this guy? I've been stuck on this for hours.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
13
votes
2answers
253 views

Crazy $\int_0^\infty{_3F_2}\left(\begin{array}c\tfrac58,\tfrac58,\tfrac98\\\tfrac12,\tfrac{13}8\end{array}\middle|\ {-x}\right)^2\frac{dx}{\sqrt x}$

Is there any chance to find a closed form for this integral? $$I=\int_0^\infty{_3F_2}\left(\begin{array}c\tfrac58,\tfrac58,\tfrac98\\\tfrac12,\tfrac{13}8\end{array}\middle|\ ...
13
votes
1answer
484 views

Tricky Integral

Can one show that $$ \int_{0}^{\infty} \! \frac {1}{1+x^{2}} \frac {x^{a}-x^{b}}{(1+x^{a})(1+x^{b})}~\mathrm{d}x=0 ~~~~~~~~ \forall ~a,b~\in \mathbb{R}. $$ Any hints?
13
votes
4answers
885 views

Good book on evaluating difficult definite integrals (without elementary antiderivatives)?

I am very interested in evaluating difficult definite integrals without elementary antiderivatives by manipulating the integral somehow (e.g. contour integration, interchanging order of ...
13
votes
2answers
243 views

Closed form of $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx,n\in\mathbb{N}$$ By Mathematica I saw that $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
13
votes
1answer
2k views

Will moving differentiation from inside, to outside an integral, change the result?

I'm interested in the potential of such a technique. I got the idea from Moron's answer to this question, which uses the technique of differentiation under the integral. Now, I'd like to consider ...
13
votes
3answers
247 views

Proving a trig infinite sum using integration

How can I prove the following using integration and elementary functions? Prove that: $$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$ $0 < \theta < 2\pi$
13
votes
3answers
335 views

For which $n$ is $ \int \limits_0^{2\pi} \prod \limits_{k=1}^n \cos(k x)\,dx $ non-zero?

I can verify easily that for $n=1$ and $2$ it's $0$, $3$ and $4$ nonzero, $4$ and $5$ $0$, etc. but it seems like there must be something deeper here (or at least a trick).
13
votes
4answers
590 views

Evaluating $\int_0^\infty\frac{\sin(x)}{x^2+1}\, dx$

I have seen $$\int_0^\infty \frac{\cos(x)}{x^2+1} \, dx=\frac{\pi}{2e}$$ evaluated in various ways. It's rather popular when studying CA. But, what about $$\int_0^\infty \frac{\sin(x)}{x^2+1} \, ...
13
votes
2answers
2k views

Average IQ of Mensa

I was wondering, what the average IQ at Mensa is. Mensa is a group of people with an IQ of at least 130. And the IQ is normally distribed with $\mu = 100$ and $\sigma = 15$. My idea was this: To ...
13
votes
3answers
419 views

Can this integral $\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$ be written in the form of a elliptic integral

I am trying to find the magnetic field due to an elliptic loop of wire. How to do integrals of the type $$\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$$ Where a and b are ...
13
votes
2answers
279 views

A new constant?

I was experimenting in Wolfram Alpha the answer to the equation $\int_0^k x^x dx=1$ And I got about 1.19... But, What is this number k (and could you calculate it to more decimal places?) And is it ...
13
votes
4answers
456 views

Evaluating $\lim \limits_{n\to \infty} \left( n \int_{0}^{\frac \pi 2} 1-\sqrt [n]{\sin x} \,\mathrm dx \right)$

Evaluate the following limit: $$\lim \limits_{n\to \infty}\,\,\, n\!\! \int_{0}^{\pi/2}\!\! \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx $$ I have done the problem . How I solved is First I ...