Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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6
votes
2answers
122 views

Integrating $\frac{x^3}{\exp(x)-1}$ from $0$ to $\infty$

While doing Physics and trying to prove the law of Stefan-Boltzmann from Plancks-law one comes to the integral \[ \int_0^\infty \frac{x^3}{\exp(x)-1} \mathrm{d}x=\frac{\pi^4}{15} \] and I would like ...
14
votes
1answer
346 views

Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$

How can we prove that: $$\int_0^1\frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx=\frac{\pi^2}{8}\zeta(3)-\frac{27}{16}\zeta(5) $$
1
vote
3answers
84 views

limit of an integral question

Let $f : [0, \infty) \to \Bbb R$ be bounded and continuous. Prove that $\lim \limits _{h \to \infty} h \int \limits _0 ^\infty e ^{-hx} f(x) \, d x = f(0)$. Our intuition was to use l'Hospital's rule ...
-2
votes
0answers
53 views

Integral calculation question [duplicate]

Calculate the following integral: $\int \limits _0 ^\frac \pi 2 \ln (\sin x) \Bbb d x$. We used the substitution $x=2t$ and then used the identity $\sin 2t = 2 \sin t \cos t$ but now we're stuck. ...
1
vote
2answers
27 views

Improper integral convergence question

Prove that the following integral converges: We divided the integral to 2 integrals (one from 0 to 1/2 and the other from 1/2 to 1). We managed to prove that the integral from 1/2 to 1 converges ...
0
votes
3answers
41 views

Integral from minus infinity to plus infinity [duplicate]

I have come across this formula through one of the members of this forum but i dont know how to prove this formula,can someone help me proving this formula. ...
0
votes
1answer
52 views

Integral of reciprocal absolute value function

I'm having issues with the integral $$\int_{-1}^1 \frac{1}{|x|}dx$$ Solving it conventionally gives me values such as $\ln 0$ and $\ln(-1)$ which are indeterminate on the real plane. Is there a way to ...
1
vote
0answers
26 views

Remainder Estimate for Integral test

I have the following question, it is a fill in the blank type question, however when I submit my answer, the system which verifies it say it is incorrect. I believe I am right, so I was hoping for ...
2
votes
4answers
44 views

Use comparison test to determine convergence

$$\int_{1}^{\infty}\frac{\ln x}{\sinh x}dx$$ I tried several functions and failed to get integrable convergent bigger function. Thanks for help.
1
vote
0answers
22 views

Continuity of improper integrals

There is a theorem saying that if $f:[a,b]\to \mathbb R$ is integrable on $[a,b]$, then $F(x):=\int_{a}^{x}f(t)dt$ is continuous on $[a,b], x \in [a,b]$. Is there an analogous theorem of the kind: ...
10
votes
4answers
416 views

What are other methods to evaluate $\int_0^1 \sqrt{-\ln x} \ \mathrm dx$

$$\int_0^1 \sqrt{-\ln x} dx$$ I'm looking for alternative methods to what I already know (method I have used below) to evaluate this Integral. $$y=-\ln x$$ $$\bbox[8pt, border:1pt solid ...
3
votes
3answers
62 views

Convergence of doubly infinite improper integral for odd functions.

I was working on this integral: $$\int_{-\infty}^{+\infty} \frac{x \, dx}{1+x^2}$$ Calculations shows that the limits DNE, and therefore the integral diverge. I used Mathematica and found the same ...
-1
votes
0answers
34 views

$\int_{1}^{\infty} \frac{\omega^2-x^2}{(\omega^2+x^2)^2}(x^2-1)^{-(2/3)}dx$

Could someone kindly evaluate $\int_{1}^{\infty} \frac{\omega^2-x^2}{(\omega^2+x^2)^2}(x^2-1)^{-(2/3)}dx$ for me? Cheers, Allen
14
votes
1answer
205 views
+50

Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$

Does $~\displaystyle{\LARGE\int}_0^1\frac{\text{arctanh }x}{\tan\bigg(\dfrac\pi2~x\bigg)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$ possess a closed form expression ? This recent ...
4
votes
7answers
128 views

Evaluating numerically $\int_0^{\infty}e^{-t^2 /100} \sin \pi t $

What is an appropriate method to approximate $$I=\int_0^\infty e^{-t^2 /100} \sin \pi t \ dt?$$ This is for a Physics problem, but in fact I need this in general, as my professor and book taught us ...
1
vote
0answers
15 views

Infinite encirclement of branch cut

Consider the integral $$I=\int _\Gamma\frac{1}{4+i(\log z)^2}dz$$ Where $\Gamma$ encircles the unit circle infinitely many times. Would it then make sense to use a parameter n: encirclement count, ...
4
votes
3answers
572 views

Prove the equation

Prove that $$\int_0^{\infty}\exp\left(-\left(x^2+\dfrac{a^2}{x^2}\right)\right)\text{d}x=\frac{e^{-2a}\sqrt{\pi}}{2}$$ Assume that the equation is true for $a=0.$
11
votes
3answers
295 views

Evaluating $\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} dx$

What is the value of the following integral? $$\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} \,dx$$ Here $\Gamma(x)$ is Euler's gamma function. EDIT: Can we improve the upper bound strictly smaller than $1$? ...
1
vote
1answer
30 views

Integration of Bessel functions:Finding a suitable contour

I have below function to integrate; $$\int_{0}^{\infty} \frac{J_{0}(ax)x^3}{k^2-x^2} dx$$ here $a,k$ are constants. Since this is an odd function, I am not allowed to extend the limits from negative ...
2
votes
1answer
51 views

integrals of exponential functions over the real axis

How to evaluate the integral $$ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})dx? $$ I intend to change the variable $x$ to $\tan t$ but failed... How to solve it?
0
votes
0answers
50 views

Contour integration from zero to infinity

When solving an improper integration from $0$ to $\infty$ which involves an even function, the integration limits can be extended from $-\infty$ to $\infty$. For example consider even function $f(x)$; ...
1
vote
2answers
50 views

Improper integrals - Showing convergence.

1)Show that for all $n\in\mathbb{N}$ the following is true: $\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$ for a constant $C>0$ and conclude that ...
1
vote
0answers
47 views

Integrate $\int_{-\infty}^{+\infty} \frac{1}{\sqrt{P(x)}}e^{-ax^2 - bx - c}dx$ where $P$ is a polynomial of degree $6$

From a physics problem I'm interested by a closed form of this integral : $$\int_{-\infty}^{+\infty} \frac{1}{\sqrt{P(x)}}e^{-ax^2 - bx - c} dx$$ where $P(x) = \lambda_6 x^6 + ... + \lambda_0$ I ...
2
votes
1answer
71 views

Arithmetic mean of $L^2$ function is $L^2$

I have found the following problem, to which I do not find the solution: Consider $f(x), x > 0$ a function such as $$ \int_0^\infty f^2(x) dx < \infty $$ and let $g(x) = \frac 1x \int_0^x ...
2
votes
1answer
58 views

$0<\int_0^\infty\frac{\sin t}{\ln(1+x+t)} dt<\frac{2}{\ln(1+x)}$

This is my first time posting so please excuse me if I don't follow the proper etiquette. This one is a rather hard problem that was assigned to me for my calculus 2 class. Thank you for your help! ...
2
votes
5answers
55 views

How to calculate: $ \lim \limits_{x \to 0^+} \frac{\int_{0}^{x} (e^{t^2}-1)dt}{{\int_{0}^{x^2} \sin(t)dt}} $

How do I calculate the follwing Limit: $$ \lim \limits_{x \to 0^+} \frac{\int_{0}^{x} (e^{t^2}-1)dt}{{\int_{0}^{x^2} \sin(t)dt}} $$ I have been solving an exam from my University's collection of ...
0
votes
1answer
73 views

Definite integration by induction

$U_n= \int\frac{x^n}{((x(1-x))^{0.5}}$ where $0<x<1$ Prove that $2nU_n=(2n-1)U_{n-1}$ My work I did $U_0=\pi, u_1=\pi/2$ so its true for $n=1$
11
votes
2answers
234 views

Easier ways to prove $\int_0^1 \frac{\log^2 x-2}{x^x}dx<0$

Prove that $$\int_0^1 \frac{\log^2 x-2}{x^x}dx<0$$ One way to do this is use the idea in the proof of Sophomore's dream. We have $$x^{-x}=\exp(-x\log x)=\sum_{n=0}^\infty\frac{(-1)^nx^n\log^n ...
8
votes
3answers
87 views

Is there a direct method for evaluating this integral: $\int_{0}^{2\pi}\ln^2(2\sin(\frac{x}{2}))dx$?

I stumbled upon this integral while attempting to evaluate $\sum_{n=1}^{\infty}\frac{\cos(n\theta)}{n}$. I started with the series $-\ln(1-z)=\sum_{n=1}^{\infty}\frac{z^n}{n}$, replaced z with ...
3
votes
4answers
523 views

Evalulate $\int_{-\infty}^{\infty}\frac{1}{(1+x^{2n})^2}dx$ by using residue theorem

I know the answer of the integral $$\int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx=\frac{\pi}{n\sin\left(\frac{\pi}{2n}\right)}$$where $n\in\mathbb{N}$. But how to evalulate ...
2
votes
1answer
45 views

Finding the value of an integral containing $(\ln x)^2$ in the denominator

While reviewing (as an instructor/test editor) a second semester calculus exam, I came across the following problem: Find the volume of the solid created by revolving around the $x$-axis the area ...
1
vote
3answers
74 views

What is wrong with my method of computing $\int_0^{\infty}x^2 e^{-x^2} \space dx$

I want to check if the following improper integral converges or diverges: $$\int_0^{\infty} x^2e^{-x^2}\space dx$$ Rewriting the integrand: $$\int_0^{\infty}-\frac{1}{2}x(-2x e^{-x^2}) \space dx$$ ...
4
votes
1answer
89 views

Exchanging the order of integration in $ \int_0^\infty \int_{-\infty}^\infty \sin(x^2)x e^{-t^2 x^2} dt dx$?

For context, this gives one way to evaluate the Fresnel sine integral at infinity. The problem I'm running into is $$ \int_0^\infty \left[ \int_{-\infty}^\infty \vert\sin(x^2)x e^{-t^2 x^2}\vert dt ...
3
votes
1answer
113 views

Help with a limit of an integral

I'm not sure how to handle limits and integral and I would like some help with the following one: let $f:[0,\infty)\rightarrow \Bbb{R}$ be a continuous and bounded function, show that ...
6
votes
3answers
94 views

Integrating $\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}$ with respect to $\theta$

I'm having some issues with the following integral $$\int_{\frac{-\pi}{2}}^\frac{\pi}{2}\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}d\theta$$ My attempt is as follows, substitute ...
10
votes
2answers
130 views

Computing $\lim_{\epsilon \rightarrow 0} \int_0^\infty \frac{\sin x}{x} \arctan{\frac{x}{\epsilon}}dx$

I'm not exactly sure how to get started computing the limit of the improper Riemann integral $$\lim_{\epsilon \rightarrow 0} \int_0^\infty \frac{\sin x}{x} \arctan\left(\frac{x}{\epsilon}\right)dx.$$ ...
0
votes
0answers
82 views

How can I resolve this improper integral?

I would like to resolve this integral numerically . However, I'm not sure about the best way to do it because it is an improper integral: $$ ...
5
votes
3answers
143 views

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

How do I evaluate this integral if I suppose that $a > 0$ $$\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ \mathrm{d}x .$$ For $a=2$ I got $2\pi$ I think the result will be ...
2
votes
0answers
98 views

A difficult integral $\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$

Here is an integral that I want to see a different approach: $$\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$$ Well, for someone who is deeply aware of the exponential integral function and the ...
0
votes
3answers
80 views

what is the integral $\int_{0}^1 \sqrt{(1+(1/x^2))}dx$

Can this integral can be evaluated? When I attempted it I got an answer of infinity. Maybe I don't know the correct method of solving this integration problem. Please provide the correct answer with ...
1
vote
0answers
21 views

Leibniz rule for probability distribution with infinite support.

Let $f$ be the pdf of a non-negative random variable $X$ with finite moments of all orders, i.e. $E[X^n]<+\infty$ for all $n \in \mathbb N$. May I apply Leibniz's rule and infer that $$\frac{d}{d ...
2
votes
2answers
67 views

HowTo solve this integral involving logarithm

I would like to solve integrals of the form $$I(c) := \int_0^\infty \log(1+x) x^{-c} \, dx ,$$ where $c \in (1,2)$. Mathematica says either 1) $I(c) = \frac{\pi}{1-c} \csc(\pi c)$ or 2) $I(c) = ...
1
vote
1answer
35 views

Improper integral existence exercise

$$\int _0^1\frac{\ln (1+\sqrt{x})}{\sin(x)} \, dx\:$$ So the singularity point is at $0$, so we`ll use this test: $$\lim _{x\to 0}\frac{\frac{\ln(1+\sqrt{x})}{\sin(x)}}{\frac{1}{\sqrt{x}}}=\lim_{x\to ...
0
votes
0answers
21 views

Compute asymptotic expansion of an integral along the unit circle

I want to compute the asymptotic expansion of the following integral with $t\rightarrow +\infty$ $\int_C\dfrac{(1+u)^{t+4}}{u^5}du$ where $C$ is the unit circle. I really appreciate your help. By ...
9
votes
3answers
227 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First I try the substitution $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
1
vote
0answers
39 views

How to calculate this Ei(x)-involved definite integral?

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al). Finally, I decided to solve this integral by ...
4
votes
1answer
44 views

Find the Principal Value of the integral $\int_{-\infty}^\infty \frac{x sin(x)}{x^2+2x+2}dx$

This problem comes from a preliminary exam from 2009 "Find the Principal Value of the integral $$\int_{-\infty}^\infty \frac{x \sin(x)}{x^2+2x+2}dx"$$ My attempt at solution: Letting $f(z)=\frac{z ...
2
votes
2answers
51 views

Calculate integral $\int_0^\infty e^{-x} (e^{-\frac a b x} - 1)^{b} dx$ for $b>0$ and any $a \in \mathbb{R}$

i am working on following task: Choose any nonzero $a \in \mathbb{R}$ so the integral converges and for a given $b > 0$ compute $\int_t^\infty e^{-x} (e^{-\frac a b x} - 1)^{b} dx$. I am looking ...
6
votes
3answers
152 views

How to integrate $\int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx$ where $a,b > 0$.

This $$\ \int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx \text{ where } a,b > 0. $$ is a problem that showed up on a GRE practice test. I believe you're supposed to use complex ...
5
votes
3answers
108 views

An improper integral with parameter $\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$

I have to analyse the convergence of this integral: $$\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$$ where $p\in \mathbb{R}$. I have thought to write: ...