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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8
votes
2answers
296 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
42
votes
3answers
1k views

How much does symbolic integration mean to mathematics?

(Before reading, I apologize for my poor English ability.) I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast ...
17
votes
7answers
949 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 $$\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx.$$ ...
12
votes
3answers
424 views

integral of $\int \limits_{0}^{\infty}\frac {\sin (x^n)} {x^n}dx$

what is the answer of $$\int \limits_{0}^{\infty}\frac {\sin (x^n)} {x^n}dx$$ I saw the answer of $$\int \limits_{0}^{\infty}\left(\frac {\sin x} {x}\right)^ndx$$ but for my question i didn't see ...
15
votes
2answers
521 views

Evaluating $\int_{-1}^{1}\frac{\arctan{x}}{1+x}\ln{\left(\frac{1+x^2}{2}\right)}dx$

This is a nice problem. I am trying to use nice methods to solve this integral, But I failed. $$\int_{-1}^{1}\dfrac{\arctan{x}}{1+x}\ln{\left(\dfrac{1+x^2}{2}\right)}dx, $$ where ...
13
votes
5answers
592 views

Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$

The problem is : Evaluate the integral $$\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$$ I have tried expand $\frac{1}{\cosh{ax}}$ and give the result in the following way: First, note ...
20
votes
4answers
2k views

Using Integration By Parts results in 0 = 1

I've run into a strange situation while trying to apply Integration By Parts, and I can't seem to come up with an explanation. I start with the following equation: $$\int \frac{1}{f} \frac{df}{dx} ...
15
votes
8answers
1k views

$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$ Evaluate Integral

Compute the following integral: $$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$$ Any hint, suggestion is welcome. Thanks.
12
votes
4answers
604 views

Computing $ \int_a^b \frac{x^p}{\sqrt{(x-a)(b-x)}} \mathrm dx$

I would like to compute the integral: $$ \int_a^b \frac{x^p}{\sqrt{(x-a)(b-x)}} \mathrm dx$$ where $$ a<b$$ and $$ p\in\mathbb{N}$$ $$ \int_a^b \frac{x^p}{\sqrt{(x-a)(b-x)}} \mathrm ...
11
votes
4answers
418 views

How to find $\int_{0}^{1}\dfrac{\ln^2{x}\ln^2{(1-x)}}{2-x}dx$

How to find $$ I=\int_{0}^{1}{\ln^{2}\left(x\right)\ln^{2}\left(1 - x\right) \over 2 - x} \,{\rm d}x $$ My idea: Let $x=1-t$, then $$ I =\int_{0}^{1}{\ln^{2}\left(1 - ...
8
votes
3answers
359 views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...
19
votes
5answers
751 views

Computing $\lim\limits_{n\to+\infty}n\int_{0}^{\pi/2}xf(x)\cos ^n xdx$

I got stuck at the following problem. Let $f\in C([0,\pi/2])$, then compute $$ \lim_{n\to+\infty}n\int\limits_{0}^{\pi/2}xf(x)\cos ^n xdx $$ Could you suggest a helpful idea?
13
votes
2answers
438 views

A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$$ I've tried substitution $y={1}/{x}$ and $e^y={1}/{x}$, but those didn't help ...
4
votes
4answers
270 views

Evaluating $\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$

How would you solve the following $$\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$$ I might be able to relate the integral to Euler sums .
12
votes
1answer
201 views

Proving that $\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$

How can we prove, without employing the aid of residues or various transforms, that, for $n>2$ $$\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$$ Motivation: ...
12
votes
5answers
19k views

Why does $1/x$ diverge?

So for the formula $\dfrac {1}{x}$, If you were to add up all $y$ values from $x=1$ to $x=∞$, wouldn't the sum approach a number because even though you are always adding, aren't you just adding ...
11
votes
4answers
423 views

Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$

Hi I am trying to find out for what values of the real parameter does the integral $$ I=\int_0^\infty \frac{\sin x}{x^s}dx $$ (a) convergent and (b) absolutely convergent. I know that the integral ...
10
votes
4answers
364 views

How to prove $f\equiv 0$ without Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know that it can be proved using Weierstrass Theorem, ...
9
votes
5answers
27k views

How to solve this integral: $\int_{-\infty}^{\infty} x^2 e^{-x^2}\mathrm dx$

I don't know how to solve it. I know there is one method using the gamma function. BUT I want to know the solution using a calculus method like polar coordinates. $$\int_{-\infty}^\infty x^2 ...
8
votes
2answers
370 views

How to evaluate this integral?

How would I prove that$$\int_{0}^\infty \frac{\cos (3x)}{x^2+4}dx= \frac{\pi}{4e^6}$$ I changed it to $$\int_{0}^\infty \frac{\cos (3z)}{(z+2i)(z-2i)}dz$$, and so the two singularities are $2i$ and ...
6
votes
4answers
368 views

How to compute $\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$

$$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$ WolframAlpha gives a numerical answer of $43.8122$, which appears to be $\sqrt{611\pi}$. And playing with that, it seems ...
4
votes
2answers
178 views

Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $

I am trying to solve this integral $$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $$ A closed form does exist despite the looks of the integrand. ...
11
votes
3answers
635 views

Fourier series of function $f(x)=0$ if $-\pi<x<0$ and $f(x)=\sin(x)$ if $0<x<\pi$

$$f(x) = \begin{cases}0 & \text{if }-\pi<x<0, \\ \sin(x) & \text{if }0<x<\pi. \end{cases}$$ My attempt: I went the route of expanding this function with a complex Fourier series. ...
6
votes
1answer
150 views

Evaluate: $I=\int\limits_{0}^{\frac{\pi}{2}}\ln\frac{(1+\sin x)^{1+\cos x}}{1+\cos x}dx$

Evaluate: $$I=\int\limits_{0}^{\frac\pi2}\ln\frac{(1+\sin x)^{1+\cos x}}{1+\cos x}dx$$
4
votes
3answers
317 views

Computing $\lim_{n \rightarrow\infty} \int_{a}^{b}\left ( f(x)\left | \sin(nx) \right | \right )$ with $f$ continuous on $[a,b]$

Let $a,b \in \mathbb{R}$ and $\textit{f} :[a,b] \rightarrow \mathbb{R}$ continuous on $[a,b]$. Does the sequence $\left (\int_{a}^{b} f(x)\left |\sin(nx) \right |dx \right )$ converge? If it does, ...
9
votes
1answer
283 views

Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx $

Prove the following $$\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx = -3\zeta(5)+\pi^2 \frac{\zeta(3)}{3}$$ where $$\operatorname{Li}^2_2(x) =\left(\int^x_0 \frac{\log(1-t)}{t}\,dt \right)^2$$ ...
9
votes
2answers
3k views

When can the order of limit and integral be exchanged?

I was wondering for a real-valued function with two real variables, if there are some theorems/conclusions that can be used to decide the exchangeability of the order of taking limit wrt one ...
8
votes
1answer
332 views

Prove $f(x)=\int\frac{e^x}{x}\mathrm dx$ is not an elementary function

How do I prove that the exponential integral $$f(x)=\int \frac{e^x}{x}\mathrm dx$$ is not an elementary function? Also, what are the general methods and tricks to prove that an integral or solution ...
6
votes
3answers
306 views

Compute $\int_0^\infty \frac{dx}{1+x^3}$

Problem Compute $$\displaystyle \int_0^\infty \frac{dx}{1+x^3}.$$ Solution I do partial fractions $$\frac{1}{x^3+1}= \frac{2-x}{3 \left( x^{2}-x+1 \right)}+\frac{1}{3 \left( x+1 \right)}.$$ But we ...
6
votes
1answer
5k views

Improper integral sin(x)/x converges absolutely, conditionaly or diverges?

$$\int_1^{\infty}\frac{\sin x}{x}dx$$ $$u=\frac{1}{x}$$ $$du=-\frac{1}{x^2}dx$$ $$dv=\sin xdx$$ $$v=-\cos x$$ $$\int_1^{\infty}\frac{\sin x}{x}dx=\frac{1}{x}(-\cos x)-\int_1^{\infty}\frac{\cos ...
6
votes
1answer
328 views

Euler's summation by parts formula

I'm beginning analytic number theory and I see this formula in Apostol's book : If $f$ has a continuous derivative $f'$ on the interval $[y,x]$, where $0 < y < x$, then $$ \sum_{y < n \le x} ...
6
votes
3answers
171 views

A problem about parametric integral

How to solve the following integral. $I(\theta) = \int_0^{\pi}\ln(1+\theta \cos x)dx$ where $|\theta|<1$
6
votes
1answer
466 views

Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...
6
votes
1answer
388 views

Functions defined by integrals (problem 10.23 from Apostol's Mathematical Analysis)

There's a problem (#10.23) in Apostol's Mathematical Analysis of which I am having a rough time solving: Let $F(y)= \int_{0}^{\infty}\frac{\sin xy}{x(x^{2}+1)}dx$ if $y > 0$. Show that $F$ ...
5
votes
3answers
717 views

On the origins of the (Weierstrass) Tangent half-angle substitution

The Weierstrass substitution is great for transforming complex trig integrals into simpler rational functions. Wikipedia suggests that it wasn't invented by Weierstrass, since Euler was already ...
3
votes
2answers
422 views

Series of nested integrals

I'm trying to calculate the following series of nested integrals with $\varepsilon(t)$ being a real function. $$\sigma = 1 + \int_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\mathrm dt_1 ...
2
votes
4answers
976 views

Approximating $\pi$ using Monte Carlo integration

I need to estimate $\pi$ using the following integration: $$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$ using monte carlo Any help would be greatly appreciated, please note that I'm a student trying to ...
12
votes
5answers
345 views

Evaluate $\int \frac{\sec^{2}{x}}{(\sec{x} + \tan{x})^{5/2}} \ \mathrm dx$

I am unable to solve the following integral: $$\int \frac{\sec^{2}{x}}{(\sec{x} + \tan{x})^{5/2}} \ \mathrm dx.$$ Tried putting $t=\tan{x}$ so that numerator has $\sec^{2}{x}$ but that doesn't help. ...
12
votes
2answers
422 views

Best way to integrate $ \int_0^\infty \frac{e^{-at} - e^{-bt}}{t} \text{d}t $

Today I had an exam and I mixed up the integration by parts formula. The question was to integrate $$ \int_0^\infty \frac{e^{-at} - e^{-bt}}{t} \text{d}t $$ I will try solve this again with the ...
10
votes
1answer
4k views

How to integrate $\int\frac{1}{\sqrt{1+x^3}}\mathrm dx$?

In a course, my teacher told us that the following integral is convergent and used the comparison test to prove it; my question is how to find the antiderivative in closed form? It seems to exist; if, ...
6
votes
3answers
2k views

Riemann integrable function

Show that $f(x)$ defined by $f(x)=\frac{1}{n}$, if $\frac{1}{n+1}<x<\frac{1}{n}, n=1,2,3,...$ and $f(0)=0$ is Riemann integrable on $[0,1]$. Also show that $$\int_0^1f(x) dx=\frac{\pi ^2}{6}-1$$ ...
6
votes
1answer
754 views

Is integration by substitution a special case of Radon–Nikodym theorem?

I was wondering if Integration by substitution is a method only for Riemann integral? if Integration by substitution is a special case of Radon–Nikodym theorem, and why? Thanks and regards!
6
votes
1answer
2k views

Derivative commuting over integral

Can a derivative operation commute over an integral operation irrespective of the properties of the function under the integral ?
5
votes
3answers
2k views

If a function $f(x)$ is Riemann integrable on $[a,b]$, is $f(x)$ bounded on $[a,b]$?

Most statements regarding Riemann integrals (at least the ones that I have encountered) begin with the statement "for $f(x)$ bounded on $[a,b]$." I am wondering if Riemann integrability implies ...
5
votes
4answers
321 views

Evaluating $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$

Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$? I've calculated that the recurrence relation for this integral is: $\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} ...
4
votes
2answers
467 views

List of functions not integrable in elementary terms

When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ...
4
votes
3answers
618 views

On the integral $\int_{-\infty}^\infty e^{-(x-ti)^2} dx$

It is known that $$ \int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}.$$ What about $$\int_{-\infty}^\infty e^{-(x-ti)^2} dx, $$ where $t \in \mathbb{R}$, $i= \sqrt{-1}$. Thanks.
3
votes
2answers
139 views

What is a simple form of this integral?

This integral reminds me of something familiar but I cannot remember the rule to make it simple. $$\int_{-\infty}^{+\infty} \frac{\exp(i a \cdot v)}v \mathrm d v$$ where $a$ is a scalar for ...
2
votes
1answer
192 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
2
votes
0answers
118 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...