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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13
votes
2answers
401 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 ...
11
votes
4answers
403 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 - ...
7
votes
4answers
217 views

How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
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
5answers
18k 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
412 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 ...
9
votes
5answers
26k 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
496 views

Reinventing The Wheel - Part 1: The Riemann Integral [closed]

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
8
votes
2answers
363 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 ...
11
votes
3answers
622 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. ...
10
votes
1answer
191 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: ...
10
votes
4answers
363 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, ...
6
votes
1answer
148 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
2answers
164 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. ...
9
votes
1answer
274 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
302 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
321 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
446 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
384 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
687 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 ...
4
votes
3answers
315 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, ...
3
votes
2answers
419 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
971 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
338 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
421 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
740 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
4answers
319 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
426 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
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 ...
4
votes
3answers
601 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
138 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
182 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
116 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 ...
1
vote
2answers
229 views

$y''=(y')^{3} e^{y}$, some easy way to solve this non-linear differential equation?

The last time when I thought that the task was about solving a non-linear differential equation with convolution and Frobenius -method more here, my instructor cheered me up that the goal was some ...
11
votes
6answers
234 views
11
votes
5answers
526 views

How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$

How can I evaluate $$I=\int_{0}^{+\infty}\!e^{\left(-ax^2-\frac b{x^2}\right)}\,dx$$ for $a,b>0$? My methods: Let $a,b > 0$ and let $$I(b)=\int_{0}^{+\infty}e^{\left(-ax^2-\frac ...
11
votes
6answers
562 views

How is the Integral of $\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$

Can Some one tell me what this method is called and how it works With a detailed proof $$\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$$ I've been using this a lot in definite integration but haven't seemed ...
7
votes
3answers
166 views

Evaluating $\int \frac{\operatorname dx}{x\log x}$

How to integrate $\frac{1}{x\log x}$? Could you give me some ideas on how to integrate this? thanks. i've tried setting $u=(\log x)^{-1}$. $\dfrac{\mathrm du}{\mathrm dx} = x^{-1}$ But it didnt ...
6
votes
0answers
290 views

Finding $\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$ if it exists [duplicate]

Does there exist the following limitation? If the answer is yes, could you show me how to find that? $$\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$$ In the following, I'm going to write what ...
6
votes
4answers
4k views

Integral of Periodic Function Question

I am stuck on a question that involves the intergral of a periodic function. The question is phrased as follows: A function is periodic with period $a$, if $f(x)=f(x+a)$ for all $x$. Question ...
5
votes
3answers
755 views

Using Residue theorem to evaluate $ \int_0^\pi \sin^{2n}\theta\, d\theta $

can you please guide me on evaluating this integral using residue theorem and binomial theorem $$ \int_0^\pi \sin^{2n}\theta\, d\theta $$ for $n = 1,2,3$ Honestly, I do not even know where to start, ...
4
votes
5answers
3k views

Derivative of Integral

I'm having a little trouble with the following problem: Calculate $F'(x)$: $F(x)=\int_{1}^{x^{2}}(t-\sin^{2}t) dt$ It says we have to use substitution but I don't see why the answer can't just be: ...
3
votes
1answer
155 views

Parseval's Identity (Integral)

Calculate the integral: \begin{equation} \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^{n}}e^{inx}\right|^{2}dx\end{equation} I'm familiar with Parseval's identity which states that for ...