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 ...

learn more… | top users | synonyms (2)

6
votes
4answers
3k 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 ...
3
votes
5answers
317 views

Prove that $\Gamma(p)\times \Gamma(1-p)=\frac{\pi}{\sin (p\pi)},\: \forall p \in (0,\: 1)$

Prove that $$\Gamma(p)\times \Gamma(1-p)=\frac{\pi}{\sin (p\pi)},\: \forall p \in (0,\: 1)$$ With $$\Gamma (p)=\int_{0}^{\infty} x^{p-1} e^{-x}dx$$ My tried: We have $$B(p, q)=\int_{0}^{1} ...
3
votes
1answer
140 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 ...
3
votes
2answers
516 views

Gaussian integrals over a half-space

Edit: I shall try to reformulate my question in order to make it -hopefully- more clear. Let $X$ be a random variable that follows the $n$-dimensional Gaussian distribution. The probability density ...
3
votes
2answers
309 views

To determine whether the integral $\int_0^{\infty} \frac{\sin{(ax+b)}}{x^p} \,\mathrm dx$ converges for $p>0$

If $p >0$, determine if the following integral converges: $$\int_1^{\infty} \frac{\sin{(ax+b)}}{x^p} \,\mathrm dx$$ What i did so far is: If $p>1$, then $$f(x)= ...
3
votes
1answer
224 views

Supremum equal limit of sequence of integrals

Show that $\sup_{x\in [a,b]} f(x) = \lim_{n\rightarrow\infty} (\int_a^b (f(x))^n \;dx)^{\frac{1}{n}}$, for f continuous and positive on [a,b]. I can show that LHS is greater than or equal to RHS but ...
1
vote
2answers
382 views

Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus

I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ...
11
votes
6answers
211 views
7
votes
2answers
208 views

Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$

$$\int_0^1 \frac{\arcsin(x)}{x}dx$$ This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma ...
5
votes
4answers
1k views

If $f^2$ is Riemann Integrable is $f$ always Riemann Integrable?

Problem: Suppose that $f$ is a bounded, real-valued function on $[a,b]$ such that $f^2\in R$ (i.e. it is Riemann-Integrable). Must it be the case that $f\in R$ ? Thoughts: I think that this is ...
4
votes
1answer
149 views

Difficult Gaussian Integral Involving Two Trig Functions in the Exponent: Any Help?

Here's the integral: $$\int_d^e \exp\left(-a\left((b+c)\cos(x)-\sqrt{b^2 - (b+c)^2 \sin^2(x)}\right)^2 \right) \, dx$$ I've tried using Mathematica: it fails. Can anyone help evaluate it? ...
4
votes
3answers
489 views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say ...
3
votes
4answers
163 views

$\int_0^{\pi}{x \over{a^2\cos^2 x+b^2\sin^2 x}}dx$

How to solve $$\int_0^{\pi}{x \over{a^2\cos^2 x+b^2\sin^2 x}}dx$$ The answer is ${\pi}^2\over{2ab}$ but I cant prove it.
3
votes
2answers
262 views

Indefinite intergral of $\int { \sqrt{ x^2-a^2} \over x } dx $

I need to integrate $$\int { \sqrt{x^2-a^2} \over x } dx $$ using substition, and show it equals $$ \sqrt{x^2 - a^2} - a(\operatorname{arcsec} ({x\over a })) +c $$ I've tried $x=a\sin t$ but I ...
3
votes
2answers
63 views

Proof about boundedness of $\rm Si$

$\def\Si{{\rm Si}}$ I want to prove the boundedness of $$\Si(x) := \int_0^x \frac {\sin \xi} \xi d\xi$$ as part of a homework (about the non-surjectivity of $\mathcal F : L^1(\mathbb R) \to ...
3
votes
3answers
2k views

Some way to integrate $\sin(x^2)$?

Because the straight forward approach involves Fresnel integrals I thought about a different approach of taking the imaginary part of $\int_{-\infty}^{\infty}\exp(ix^2) $ but have no idea how to ...
3
votes
2answers
346 views

Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables

Problem 3-29 (p. 61) in the section treating Fubini´s theorem reads: Use Fubini´s theorem to derive an expression for the volume of a set of $\mathbb{R}^{3}$ obtained by revolving a Jordan-measurable ...
3
votes
6answers
321 views

Integration of $\int\frac{1}{x^{4}+1}dx$. [duplicate]

I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}dx$. Do I have to use trigonometric substitution?
2
votes
2answers
607 views

If a function is integrable, then it is bounded

Probably a simple question, but I wonder about the following. I know that if a function $f : \mathbb{R} \rightarrow \mathbb{R} $ is (Riemann)integrable, then it is bounded. I wonder if I can ...
2
votes
3answers
112 views

Fractional Trigonometric Integrands

$$∫\frac{a\sin x+b\cos x+c}{d\sin x+e\cos x+f}dx$$ $$∫\frac{a\sin x+b\cos x}{c\sin x+d\cos x}dx$$ $$∫\frac{dx}{a\sin x+\cos x}$$ What are the relations between the numerator in the denominator, and ...
2
votes
1answer
188 views

Is this definition missing some assumptions?

In his "Calculus on manifolds" Spivak first defines $n$-dimensional (Riemann-) integral over rectancles, then over Jordan measurable subsets of rectangles and finally extends it to open sets using ...
2
votes
2answers
291 views

Convergence of Lebesgue integrals

I am sitting on this multiple-choice question and I cannot answer it, nor say if it is right or wrong: Given non-negative, Lebesgue-integrable functions $f,f_k\colon E\rightarrow \mathbb{R}^+$ with ...
1
vote
1answer
7k views

Integral of sqrt{1-x^2} using Integration by parts

I was asked to solve this indefinite integral using Integration by parts. $$\int \sqrt{1-x^2} dx$$ I know how to solve if use the substitution $x=sin(t)$ but I'm looking for the Integration by parts ...
0
votes
1answer
55 views

About the confluent versions of Appell Hypergeometric Function and Lauricella Functions

I know there are two important properties about Appell Hypergeometric Function and Lauricella Functions: ...
0
votes
4answers
611 views

Calculate $\int\left( \sqrt{\tan x}+\sqrt{\cot x}\right)dx$ [duplicate]

How to calculate following integration? $$\int\left( \sqrt{\tan x}+\sqrt{\cot x}\right)dx$$
109
votes
1answer
6k views

Evaluate $ \int_{0}^{\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan x)}\,\mathrm dx$

Evaluate the following integral$$\int_{0}^{\Large\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan x)}\,\mathrm dx$$ My Attempt: Let $$\tag1 I = \int_{0}^{\Large\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan ...
28
votes
2answers
2k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
27
votes
1answer
743 views

Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}dx=\frac{\pi^2}8-\frac12$

How can I prove the following identity? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}dx=\frac{\pi^2}8-\frac12$$
23
votes
2answers
406 views

A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

Is it possible to evaluate the following integral in a closed form? $$\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx,$$ where $\phi$ is the golden ratio: ...
23
votes
3answers
654 views

A conjectured closed form of $\int_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

Consider the following integral: $$\mathcal{I}=\int_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx.$$ I tried to evaluate $\mathcal{I}$ in a closed form (both manually and using ...
23
votes
2answers
738 views

Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?

This post of Boris Bukh mentions amazing Gustav Herglotz's integral $$\int_0^1\frac{\ln\left(1+t^{\,4\,+\,\sqrt{\vphantom{\large A}\,15\,}\,}\right)}{1+t}\ \mathrm ...
23
votes
1answer
482 views

$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$

I need help with calculating this integral: $$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$ where ...
45
votes
13answers
6k views

Why do we need to learn integration techniques?

After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math. These classes are integral calc and multivariable calc. Integral calc ...
17
votes
3answers
346 views

Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$

I'm trying to figure out how to evaluate the following: $$ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx $$ I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
14
votes
4answers
697 views

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

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ ...
27
votes
4answers
771 views

Evaluating $\int_{0}^{1}\frac{1-x}{1+x}\frac{dx}{\ln x}$

Some time ago I came across to the following integral: $$I=\int_{0}^{1}\frac{1-x}{1+x}\frac{dx}{\ln x}$$ What are the hints on how to compute this integral?
16
votes
4answers
787 views

Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx = \frac{\pi^3}{16}-3G\log 2 \tag{1}$$ where $G$ is Catalan's Constant. I was able to express it in terms of ...
13
votes
2answers
436 views

Integral $\int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma x)\,dx$

I am having trouble showing this equality is true$$ \int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma ...
12
votes
5answers
290 views

The other ways to calculate $\int_0^1\frac{\ln(1-x^2)}{x}dx$

Prove that $$\int_0^1\frac{\ln(1-x^2)}{x}dx=-\frac{\pi^2}{12}$$ without using series expansion. An easy way to calculate the above integral is using series expansion. Here is an example ...
12
votes
3answers
237 views

Compute $\int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx$

How to compute the following integral \begin{equation} \int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx \end{equation} I have been given two integral questions by my teacher. I cannot answer ...
12
votes
5answers
513 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 ...
7
votes
1answer
3k views

When can we exchange order of two limits?

My questions are about a sequence or function with several variables. I vaguely remember some while ago one of my teachers said taking limits of a sequence or function with respect to different ...
19
votes
6answers
495 views

The formalism behind integration by substitution

When you are doing an integration by substitution you do the following working. $$\begin{align*} u&=f(x)\\ \Rightarrow\frac{du}{dx}&=f^{\prime}(x)\\ \Rightarrow ...
16
votes
1answer
294 views

Prove the following integral inequality

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: ...
14
votes
3answers
1k views

Evaluating $ \int_0^{\pi/2}\frac{(\ln{\sin x})(\ln{\cos x})}{\tan x}dx $

I need to solve $$ \int_0^{\pi/2}\frac{(\ln{\sin x})(\ln{\cos x})}{\tan x}dx $$ I tried to use symmetric properties of the trigonometric functions as is commonly used to compute $$ ...
13
votes
4answers
465 views

Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$

For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$. My attempt: Let $$\begin{align*} f_n(x) &= \frac{\ln\left(1-2 ...
9
votes
1answer
363 views

Help with a troublesome double integral

I'm having difficulty with a double integral $$-2i\int_{0}^{\infty}\int_{0}^{\infty}\frac{dxdt}{t(e^{2\pi x}-1)(e^{2\pi t/s}-1)}\left[\cos(t\log(1-ix))-\cos(t\log(1+ix))\right]$$ where ...
18
votes
3answers
399 views

Closed form for integral $\int_{0}^{\pi} \left[1 - r \cos\left(\phi\right)\right]^{-n} \phi \,{\rm d}\phi$

Is there a closed form for $$I_n =\int_{0}^{\pi} \frac{\phi}{(1 - r \cos\phi)^n} \,{\rm d}\phi $$ for $\left\vert\,r\,\right\vert < 1$ real and $n > 0$ integer ? The solution to this integral ...
10
votes
2answers
325 views

Stieltjes Integral meaning.

Can anybody give a geometrical interpretation of the Stieltjes integral: $$\int_a^bf(\xi)\,d\alpha(\xi)$$ How would we calculate? $$\int_a^b \xi^3\,d\alpha(\xi)$$ for example.
8
votes
3answers
321 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 - ...