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

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
45 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 $\boldsymbol{r}\in\mathbb{R}^3\setminus\overline{V}$ be any point external to $V$. I intuitively suppose that ...
4
votes
1answer
68 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
2answers
52 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!
3
votes
1answer
41 views

Is the following integral identity true or not? [closed]

Is the following statement true or not?$$\int_{-\infty}^\infty xf(x)\,dx = \left. {d\over{dt}} \int_{-\infty}^\infty e^{tx}f(x)\,dx\right|_{t = 0}$$
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 ...
0
votes
3answers
59 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 ...
1
vote
0answers
40 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 ...
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
37 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 ...
0
votes
1answer
23 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 ...
1
vote
4answers
151 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 ...
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(...
0
votes
0answers
64 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 ...
3
votes
4answers
119 views

If $\int_2^\infty f(x)^2 dx $ is convergent, is it true that $\int_2^\infty f(x)x^{-3/4} dx $ is convergent?

If $\int_2^\infty f(x)^2 dx $ is convergent, is it true that $\int_2^\infty f(x)x^{-3/4} dx $ is convergent?
1
vote
1answer
77 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 ...
0
votes
1answer
61 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 ...
4
votes
1answer
96 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
64 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/...
1
vote
2answers
50 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{...
2
votes
3answers
75 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 ...
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,...
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
1answer
101 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\...
11
votes
7answers
439 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=\...
3
votes
1answer
51 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 ...
4
votes
1answer
74 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?...
2
votes
0answers
30 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
3answers
63 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
3answers
195 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}^{+\...
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)\...
8
votes
2answers
188 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
0answers
41 views

How to calculate improper integral?

Let $I= \int_0^\infty \frac{1-cos(t)}{t^{\alpha+1}}dt$, where $0 \lt \alpha \lt 2$. I want to calculate $I$, actually according to wolfram $I=-\cos(\frac{\alpha \pi}{2}) \Gamma(-\alpha)$, but is ...
0
votes
0answers
53 views

Compute the partial sums in a closed-form of $\sum_{n=1}^\infty\frac{e^{-nx}}{e^{nx}-1}$, with $x>0$ or a related series

One can do the change of variable $x=nv$ in the integral formula (3) here page 2 to get after summation $$\zeta(s)^2-\zeta(s)-\frac{1}{s-1}\zeta(s)=\frac{1}{\Gamma(s)}\int_0^\infty\sum_{n=1}^\infty \...
1
vote
2answers
26 views

Integral test application

I'm pretty sure I haven't made any mistakes in part 1), but I have absolutely no idea how to do part 2). I don't even know what the question is asking and I've never seen anything like it before. I ...
9
votes
2answers
251 views

integral inequality for $f(x)$ and $f(\sqrt{x})$

Show that if $f(x)\in [0;1]$, $f\in C$ and $\int\limits_{1}^{+\infty}f(t)dt=A$ then $\int\limits_{1}^{+\infty}tf(t)dt>\frac{A^2}{2}$ I only have noticed two small things: If $A=1$ inequality is ...
27
votes
2answers
603 views

A novelty integral for $\pi$

My lab friends always play a mentally challenging brain game every month to keep our mind running on all four cylinders and the last month challenge was to find a novelty expression for $\pi$. In ...
4
votes
3answers
92 views

Prove $\int_0^{\infty} \int_0^{\infty} \frac{\sqrt{xy} ~dxdy}{(x+y)(1+x y)^s}=\frac{\pi}{2(s-1)}$ for $s>1$

By evaluation with WolframAlpha for different values of $s$ it is apparent that: $$I(s)=\int_0^{\infty} \int_0^{\infty} \frac{\sqrt{xy} ~dxdy}{(x+y)(1+x y)^s}=\frac{\pi}{2(s-1)},~~~~~s>1$$ I'...
1
vote
1answer
58 views

Find the values of $a$ such that the integral converges.

We have the improper integral $$\int_0^\infty \frac{\tanh(x)}{\left(1+x^2\right)^a}\, \mathrm{d}x$$ and we need to find all the values of $a$, such that the above integral converges. We know ...
5
votes
3answers
105 views

Prove $\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2} \right)$

Wolfram Alpha evaluates this integral numerically as $$\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=0.379064 \dots$$ Its value is apparently $$\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}...
2
votes
1answer
94 views

Prove $\int_0^\infty \exp (-x^2) dx=\int_0^\infty \exp \left(-x^2 \left( 1-\frac{4}{x^2-2}\right)^2 \right) dx$

It appears by numerical evaluation that: $$\int_0^\infty \exp \left(-x^2 \left( 1-\frac{4}{x^2-2}\right)^2 \right) dx=\int_0^\infty \exp (-x^2) dx=\frac{\sqrt{\pi}}{2}$$ The plot of the difference ...
1
vote
1answer
31 views

Integrals of a function and its absolute value

Is the following proposition true? Let $f(x)$ be a real-valued function defined on $[a,b] \subset \mathbb{R}$, and suppose that the integral, $$ I = \int_a^b f(x) dx, $$ exists in the sense of ...
4
votes
1answer
97 views

Given $\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$, evaluate $\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx $

Given $$\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$$ evaluate: $$\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx. $$ I can find that $$\left(ax+\frac{b}{x}\right)^2 = a^2x^2+2ab+\frac{b^2}{x^2}...
0
votes
1answer
39 views

Quick question about improper integral

What do I do if in the point of lower bound of some first-odered improper intagral integrand doesn't exists? For instance, $$\int _1^{\infty }\frac{dx}{x\log ^2x} $$
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}^...
6
votes
0answers
189 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 ...
7
votes
3answers
181 views

Showing $\pi\int_{0}^{\infty}[1+\cosh(x\pi)]^{-n}dx={(2n-2)!!\over (2n-1)!!}\cdot{2\over 2^n}$

Showing $$\pi\int_{0}^{\infty}[1+\cosh(x\pi)]^{-n}dx={(2n-2)!!\over (2n-1)!!}\cdot{2\over 2^n}\tag1$$ Recall $$1+\cosh(x\pi)={(e^{x\pi}+1)^2\over 2e^{x\pi}}\tag2$$ $$I_n=2^n\pi\int_{0}^{\...
5
votes
3answers
204 views

Prove $\int_{0}^{\infty}{1\over x}\cdot{1-e^{-\phi{x}}\over 1+e^{\phi{x}}}dx=\ln\left({\pi\over 2}\right)$

Integrate $$I=\int_{0}^{\infty}{1\over x}\cdot{1-e^{-\phi{x}}\over 1+e^{\phi{x}}}\,dx=\ln\left({\pi\over 2}\right)\tag1,$$ where $\phi={1+\sqrt5\over 2}$. Recall $\tanh y=-{1-e^{2y}\over 1+e^{2y}}...