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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2
votes
1answer
30 views

Convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$

Problem: Analyze the convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$. It seems to me that 'I' converges for $0<a<1$. My work: I wrote integral 'I' as a ...
1
vote
4answers
146 views

How to integrate $\int _1^{\infty }\frac{dx}{\left(x^2+1\right)\sqrt{x^2-1}}= \;?$

How do I integrate $\int _1^{\infty }\left(\frac{1}{\left(x^2+1\right)\sqrt{x^2-1}}\right)\:dx$? So what I've tried is substituting $x\:=\:\frac{1}{\sin t}$. So then I'll have that when $x\rightarrow ...
36
votes
2answers
953 views
+50

Prove that $\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

Question. Let $$ f(x)=\!\left\{\,\,\, \begin{array}{ccc} \displaystyle{\left\lfloor{1\over x}\right\rfloor}^{-1}_{\hphantom{|_|}}&\text{if} & 0\lt x\le 1, \\ & \\ 0^{\hphantom{|^|}} &\...
0
votes
3answers
117 views

Value of the integral $\int_{a}^{a}f(x)dx?$

I am confused about the value of the integral $\int_{a}^{a}f(x)dx,$ is it $0$ or not always? In Riemann integration case (i.e. $f$ is bounded ) its zero by definition. What about other cases for ...
1
vote
1answer
27 views

Continuity of the improper integral $\int_{0}^{x}y^{-1/2}dy.$

The improper integral $\int_{0}^{x}y^{-1/2}dy$ is $1.$ Continuous on $[0,\infty).$ $2.$ Continuous on $(0,\infty).$ $3.$ Continuous only on $[1/2,\infty)$ It is cleat that option $3$rd is wrong ...
65
votes
7answers
4k views

Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
6
votes
2answers
104 views

Why is $\int_{-1}^{1} \frac{1}x \mathrm{d}x$ divergent?

Isn't $$\int_{-1}^{1} \frac{1}x \mathrm{d}x=\lim_{\epsilon\to 0^{+}} \int_{-1}^{-\epsilon} \frac{1}x \mathrm{d}x+\int_{-\epsilon}^{\epsilon} \frac{1}x \mathrm{d}x+\int_{\epsilon}^{1} \frac{1}x \mathrm{...
1
vote
1answer
33 views

Lebesgue integral of $\frac{1}{\|\boldsymbol{x}-\boldsymbol{r}\|^2}$ on an infinite cylinder

Let $V\subset \mathbb{R}^3$ be a solid infinite cylinder, or cylindrical shell, and let $\boldsymbol{r}\in\mathbb{R}^3$ be any point of the space. I intuitively suppose that the Lebesgue integral $$\...
3
votes
2answers
45 views

Improper integral - checking convergence of $\int_{1}^{\infty} x^2 \sin(x^4) dx$

Does the following improper integral converges ? $$\int_{1}^{\infty} x^2 \sin(x^4) dx$$ Tried to find some known improper integral to compare this one to, but didn't find one. Thanks for helping!
4
votes
1answer
67 views

Proving that a function is integrable

Given that $f:[0, \infty] \to \mathbb{R}$ is decreasing with $\displaystyle\lim_{x \rightarrow \infty} f(x)=0$, prove that $$I=\int_{0}^{1}\frac{\cos(\frac{1}{x})f(\frac{1}{x})}{x^2}dx$$ converges. ...
3
votes
1answer
41 views

Is the following integral identity true or not? [on hold]

Is the following statement true or not?$$\int_{-\infty}^\infty xf(x)\,dx = \left. {d\over{dt}} \int_{-\infty}^\infty e^{tx}f(x)\,dx\right|_{t = 0}$$
3
votes
4answers
116 views
14
votes
3answers
568 views

How to integrate $\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{{x}^{n}}}{n!}\ \text{d}x} $?

I have done one with $\displaystyle\int_0^{\infty}\frac{x-\sin x}{x^3}\ \text{d}x$, but I have no ideas with these: $$\begin{align*} I&=\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{...
1
vote
1answer
46 views

Limits at infinity of a function with convergent improper integral

Let $f:[0,\infty)\to\mathbb{R}$ be integrable in everywhere. Suppose $\int\limits_0^{\infty}|f(t)|dt$ converges. Show that there exists a sequence $x_n$ such that $x_n\to\infty$ while $f(x_n)\...
2
votes
1answer
29 views

Comparison test with improper integral

I have the integral $$\int_2^\infty\frac{3}{\sqrt[3]x(x+2\sqrt x)}dx$$ and have to find out whether it's divergent or convergent using the comparison test. I've been trying to understand this topic ...
1
vote
1answer
49 views

Inner product on $\mathbb{R}[X]$

Let $P$ and $Q$ be two polynomials in $\mathbb{R}[X]$ and let $$\langle P,Q\rangle =\int _{-\infty}^{+\infty}P(x)Q(x)f(x)dx$$ with $f(x) = \frac{1}{\sqrt{2\pi}} \exp(-x^2/2)$. I would like to ...
17
votes
3answers
340 views

Prove $\pi^2\int_0^\infty\frac{x\sin^4\pi x}{\cos\pi x+\cosh\pi x}dx=e^2\int_0^\infty\frac{x\sin^4ex}{\cos ex+\cosh ex}dx=\frac{176}{225}$

Marco Cantarini and Jack D'Aurizio proved hard-looking integrals (see Marco and Jack) in my recent two posts. This is our final hard-looking integral that yield a rational answer: $$\pi^2\int_{...
0
votes
3answers
57 views

convergence of $\int _0^1\:\frac{\ln\left(1-x\right)}{\left(1+x\right)^2}dx$

How do I prove convergence of $$\int _0^1\:\frac{\ln\left(1-x\right)}{\left(1+x\right)^2}dx$$ and if it's convergent, calculate the value of the integral? I noticed that the values that the function ...
2
votes
1answer
40 views

Where is the mistake of a possible application of Frullani's theorem in this case?

My question is about what is the problem, if there is one, to get an identity using Frullani's integral. I've in a hand the statement from MathWorld, and other statement from this site, with a nice ...
1
vote
1answer
35 views

Definite integral involving algebraic, exponential, and product of two Meijer's G function

I am having trouble with calculating the following integral: \begin{equation} I = \int_{0}^{\infty}x\exp({-\beta x})\large{G}_{2,2}^{1,2}\left( x \left| \begin{array}{cc} 1,1 \\ 1,0 \end{array} \...
1
vote
0answers
37 views

Continuous positive function such that $\int\limits_1^{\infty}f(x)dx$ converges, while $\int\limits_1^{\infty}f^2(x)dx$ diverges [duplicate]

Does there exist a continuous positive function such that $\int\limits_1^{\infty}f(x)dx$ converges, while $\int\limits_1^{\infty}f^2(x)dx$ diverges? I have proved that if $f$ is decreasing ...
1
vote
1answer
36 views

Convergence of the integral: $I_{\alpha }=\int _0^{\infty }\left(\frac{e^{-\alpha x}}{\sqrt{x}}\right)\:dx$

Study the convergence of the integral: $$I_{\alpha }=\int _0^{\infty }\left(\frac{e^{-\alpha x}}{\sqrt{x}}\right)\:dx$$ and calculate $I_2$. Ok so to study the convergence I'm using convergence ...
1
vote
2answers
47 views

Closed form and limit of the integral of a rational function

While trying to answer this question, I wondered whether there could be a way to: (A) Find the closed form of the generalization of integrals $I$ and $J$, that is $$I_n=\int_{-\infty}^{+\infty}\frac{...
0
votes
1answer
20 views

Short-time Fourier Transform identity in $L^2$

Define the Short-time (or windowed) Fourier Transform of a function $f:\mathbb{R}\rightarrow\mathbb{C}$ as follows, $F_gf(\omega,t)=\int\limits_{\mathbb{R}}f(x)\overline{g(x-t)e^{ix\omega}}dx$. Show ...
22
votes
5answers
1k views

Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$

I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
0
votes
0answers
56 views

How much can the integrability at zero tell about the decay rate around zero?

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
2
votes
2answers
38 views

for $f\in C^2(\mathbb{R})$, finding the derivative of $\frac{d}{dt}\int_0^\infty f(x+t)\cdot xdx$

Let $f\in C^2(\mathbb{R})$, (a) Prove that $$\frac{\mathrm{d}}{\mathrm{d}t}\int_0^\infty f(x+t)\cdot x\mathbb{d}x=-\int_0^\infty f(x)\mathrm{d}x$$ (b) Prove that $$ \iint_{(0,\infty)\times(...
11
votes
7answers
388 views

Prove that $\int_{0}^{\infty}{1\over x^4+x^2+1}dx=\int_{0}^{\infty}{1\over x^8+x^4+1}dx$

Let $$I=\int_{0}^{\infty}{1\over x^4+x^2+1}dx\tag1$$ $$J=\int_{0}^{\infty}{1\over x^8+x^4+1}dx\tag2$$ Prove that $I=J={\pi \over 2\sqrt3}$ Sub: $x=\tan{u}\rightarrow dx=\sec^2{u}du$ $x=\...
0
votes
1answer
58 views

Integral of Fractional Part $\int_{0}^{1} \{ \frac{1}{x} \}dx$

Does the integral exist? $\displaystyle\int_{0}^{1}\{\frac{1}{x}\}dx,\quad$ where {x} is the fractional part. I have broken it into $$\displaystyle\int_{0}^{1}\frac{1}{x}-\lfloor \frac{1}{x} \rfloor ...
1
vote
1answer
74 views

$\int_{- \infty}^{+ \infty} |f(t)| dt < \infty \implies \int_{-\infty}^{x} f(t) dt$ is continuous?

I've found counter example for $(A),(D)$ and have shown except a bounded interval $F$ is uniformly continuous everywhere else. And so $(B)$ would imply $(C)$ is correct. But I can't show $(B)$ is ...
6
votes
0answers
186 views

Juantheron-like integral

While seeing this post, the following integral is just struck me \begin{equation} \int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1 \end{equation} I have tried like what user @OlivierOloa did in ...
4
votes
1answer
95 views

Evaluate improper integral: Exponent of square root.

I was working in a problem in physics, which gave me this integral, and I need solution: $$ \int_{-\infty}^\infty \exp{\left(-\sqrt{x^2 + a^2}\right)}dx $$ The problem is, I have no clue how to start....
2
votes
1answer
60 views

evaluate if integral converge & determine antiderivative

The problem is i need to study the convergence of A and B and find the antiderivative of C $$A=\int_0^\infty \frac{\sin(x) +x}{\sqrt x + x^3}dx$$ $$B=\int_0^\infty \frac{1}{\sqrt {e^x-1}(x^2+x^{1/...
2
votes
3answers
74 views

Trying to solve improper integral

I've been trying to solve this $$ \int_{-\infty}^\infty {\sin(x)\over x+1-i }dx $$ using residue theorem. I've tried using a square contour pi, pi+pii, -pi+pii, pi and half a circle but with the ...
2
votes
1answer
98 views

For what values of $a$ does $\int_0^\infty\left(\frac{x^a}{1 + x^2}\right)^4 \, dx$ converge?

I'm learning about convergence/divergence of improper integrals and need help with the following problem: Find for what values of $a$ does the following integrals exists $$(1) \int_0^\infty\...
0
votes
2answers
63 views

proving integral equality using substitution

"Using the substitution $t=\tan \frac{x}{2}$, prove that for every $-1<r<1$, $\int_{0}^{\pi}\frac{\cos x}{1-2r\cos x+r^2}dx=\int_{0}^{\pi}\frac{r}{1-2r\cos x+r^2}dx$ " I've tried the suggestion,...
7
votes
2answers
151 views

Prove that $f(a)=f(b)$ if $\int_{-\infty}^{\infty}f(x)dx=1$

Task is: $f(x)$ is positive, continious function in the field of real numbers, and $\int_{-\infty}^{\infty}f(x)dx=1$. Let $\alpha\in(0,1)$, and length of $[a,b]$ is minimal from all $\int_{a}^{b}f(x)...
4
votes
1answer
71 views

Convergence of the integral $\int_0^\infty f(x)\frac{xf'(x/(1-1/N))}{f(x/(1-1/N))}\ \mathsf dx$ as $N\to\infty$

How can calculate this integral $$\lim_{N \to \infty} \int_{x=0}^{\infty}f(x) \frac{x f'\left(\frac{x}{1-1/N}\right)}{f\left(\frac{x}{1-1/N}\right)} dx$$ where $f(x)$ is a probability density function?...
0
votes
2answers
32 views

Convergence of an integral involving tan function

How would i prove that integral $$\int_0^{1}{\frac{\tan^2(x)}{\sqrt{x^5}}}$$ converges? By using some plotting apps, I managed to find that $\tan^2(x) \le 3x^2$ for $x \in (0, 1)$ (which would ...
2
votes
0answers
29 views

Confusion about the convergence of Riemann zeta function in terms of the integral

Titchmarsh wrote that $$\zeta(s)=s\int_{1}^{\infty}\frac{\left \lfloor x \right \rfloor-x+\frac{1}{2}}{x^{s+1}}\,\mathrm{d}x+\frac{1}{s-1}+\frac{1}{2}\tag{2.14}$$ using the Euler-Maclaurin summation, ...
4
votes
1answer
3k views

Laplace transform of the Bessel function of the first kind

I want to show that $$ \int_{0}^{\infty} J_{n}(bx) e^{-ax} \, dx = \frac{(\sqrt{a^{2}+b^{2}}-a)^{n}}{b^{n}\sqrt{a^{2}+b^{2}}}\ , \quad \ (n \in \mathbb{Z}_{\ge 0} \, , \text{Re}(a) >0 , \, b >0 ...
3
votes
1answer
48 views

Improper integral complex analysis $\int_{-\infty}^\infty \frac{e^{ax} \, dx}{\cosh(x)}$

I tried the following problem but I don't think I got the right answer. I checked it by substituting $a=\frac{1}{2}$ into the integral and putting that through Wolfram Alpha but it didn't match the ...
28
votes
5answers
1k views

Prove $\int_0^\infty \frac{\ln \tan^2 (ax)}{1+x^2}\,dx = \pi\ln \tanh(a)$

$$ \mbox{How would I prove}\quad \int_{0}^{\infty} {\ln\left(\,\tan^{2}\left(\, ax\,\right)\,\right) \over 1 + x^{2}}\,{\rm d}x =\pi \ln\left(\,\tanh\left(\,\left\vert\, a\,\right\vert\,\right)\,\...
0
votes
1answer
142 views

Contour integration when pole is outside the contour

Here they are using the pole OUTSIDE the contour? I thought this was illegal according to the residue theorem or we are not supposed to do contour integration with poles outside the contour itself.
-2
votes
1answer
42 views

Test improper integral with $\ln$ for convergence [closed]

Can you help me to test this integral for convergence, please $$\int\limits_1^e \frac{1}{\sqrt{1 - \ln^2x}}\,dx$$
3
votes
3answers
51 views

$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx $

$$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx$$ where $a>b>0$ What shall I do? I have diffucty when I meet multi value function.
0
votes
1answer
20 views

An upper bound of $ \left| \frac{1}{s}\log\zeta(s) \right| $ for $\Re s=\sigma>1$, from this integral formula and a related comparison

For $\Re s=\sigma>1$ one has the following known formula $$\frac{1}{s}\log\zeta(s)=\int_1^\infty \Pi(x)x^{-s-1}dx,$$ then if we take the derivative we can write $$\frac{1}{s}\log\zeta(s)=s(s+1)\...
7
votes
1answer
142 views

Prove that $\int_{0}^{1}{(1-x)(x-3)\over 1+x^2}\cdot{dx\over \ln{x}}={\ln{8\over \Gamma^4(3/4)}}$

Prove $$I=\int_{0}^{1}{(1-x)(x-3)\over 1+x^2}\cdot{dx\over \ln{x}}=\color{blue}{\ln{8\over \Gamma^4(3/4)}}\tag1$$ $(1-x)(x-3)=-x^2+4x-3$ $${1\over 1+x^2}=\sum_{n=0}^{\infty}(-1)^nx^{2n}\tag2$$ ...
0
votes
3answers
185 views

Evaluate the integral $ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx $ using the residue method

I have a problem in evaluating the integral above. So far I've proceeded in this way. We have an even function, so: $$ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx = \frac{1}{2} \int_{-\infty}^{+\...
1
vote
1answer
53 views

$w(x,y)=\frac{x^2+3y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dx - \frac{3x^2+y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dy$ , calculate $\int_{+\gamma} w $

$\gamma$ is the curve of this equation: $$\rho=e^{-\theta} \qquad \theta \in [0,+\infty)$$ It is oriented in the growing $\theta$ $$w(x)= \sum_{i=1}^n a_i(x) \ dx_i $$ $$\int_{+\gamma} w=\sum_{i=1}^...