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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0
votes
2answers
58 views

Finding $(p,q)$ such that $\frac{x^p}{1+x^q}$ is integrable on $(0,+\infty)$

I'm trying to show that $f(x) = \frac{x^p}{1+x^q}$ is integrable on $(0,\infty)$ if and only if $p > -1$ and $q-p > 1$. So on $[1,\infty)$ we can compare with $g(x) = x^{p-q}$ which is ...
6
votes
3answers
127 views

Finding the integral of $\frac{x}{e^x + 1}$ [duplicate]

I've having some difficulty with finding this integral: $$ \int_0 ^{\infty} \frac{x}{e^x + 1}$$ Now usually I would use the monotone convergence theorem to write (using geometric series): $$f_n (x) ...
1
vote
1answer
35 views

Which singularities are integrable

Consider a function $f: \mathbb{R}^n \to \mathbb{R}$ that has a singularity at point $x_0$ such that $\lim_{x\to x_0} f(x) = +\infty$. When we can be sure that this singularity is integrable? (you can ...
1
vote
1answer
42 views

Regularizing conditionally convergent integrals

For functions $f:\mathbb{R}\to \mathbb{R}$ the definition of a convergent improper integral is straightforward: the integral $\int_\mathbb{R} f(x)dx$ converges iff $$\lim_{(a,b)\to (-\infty, ...
1
vote
2answers
526 views

Evaluating integral using Beta and Gamma functions.

$$\int_0^{\infty} x^{-3/2} (1 - e^{-x})\, dx$$ Evaluate the above integral with the help of Beta and Gamma functions. I'm badly stuck. I'm getting Gamma of a negative number and have no clue how ...
0
votes
1answer
47 views

An improper integral question

everyone who is interested calculus, I wonder ask a question about the value of an improper integral. Here is the integral: $\int_0^\infty \! \frac{e^{-x}}{x} \, \mathrm{d}x $ Is it diverge ( how to ...
0
votes
2answers
88 views

How to show that $\int_{-\infty}^\infty\frac{t}{(a^2+t^2)(b^2+t^2)(e^{2\pi t}-1)}dt=\frac{1}{2ab(a+b)}+\frac{1}{b^2-a^2}\sum_{a<k\leq b}\frac{1}{k}$

I'm stuck on this problem. Here $a,b\in\mathbb{N}$ with $b>a$. I have already shown that $$-\lim_{\varepsilon\searrow 0}\int_{|t|>\varepsilon}\frac{\coth(\pi ...
2
votes
0answers
44 views

Difficult integrals, do they converge, show there's no dependence on parameters.

I am trying to figure out whether these integrals: a) $$\int_{\mathbb R^2}{{\rm d}\xi \over \left\vert\vphantom{\Large A}\,\log\left(\left\vert\,x - \xi\,\right\vert\right) -\log\left(\left\vert\,y ...
1
vote
2answers
90 views

Limit of an integral $\int_0^{\pi/2} \frac{x\sin^nx}{\sqrt{1+\sin^2 x}}\ dx$

What's the limit of $$\lim_{n\longrightarrow \infty} \int_0^{\pi/2} \frac{x\sin^nx}{\sqrt{1+\sin^2 x}}\ dx\ ?$$ Lebesgue's theorem is useless here I think, since $\sin x$ has no limit for ...
2
votes
1answer
72 views

Show that $\int_{-\infty}^{\infty}\frac{x^2dx}{(x^2+1)^2(x^2+2x+2)}=\frac{7\pi}{50}$

Show that $$\int_{-\infty}^{\infty}\frac{x^2dx}{(x^2+1)^2(x^2+2x+2)}=\frac{7\pi}{50} $$ So I figured since it's an improper integral I should change the limits ...
1
vote
2answers
82 views

Convergence of $\displaystyle\int\frac{1}{\sqrt[3]{1-x^3}}\ dx$

Please help me to prove that this integral converges. $$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx $$ No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result ...
1
vote
1answer
41 views

Calculating integral with different boundary

I am confused about calculting such integral with two ways of choosing boundary: $$I_1 = \int_0^1 \frac 1{x^2}dx,\quad \text{for } 0 \le x \le 1$$ $$I_2 = \int_0^1 \frac 1{x^2}dx, \quad \text{for } 0 ...
10
votes
3answers
355 views

How to evaluate improper integral $\int_{0}^{\infty}\frac{\tan^{-1}{x}}{e^{ax}-1}dx$?

I'm trying to evaluate the improper integral, $$I(a)=\int_{0}^{\infty}\frac{\tan^{-1}{x}}{e^{ax}-1}\mathrm{d}x,~~~\text{where }a\in\mathbb{R}^+.$$ Does this integral have a simple closed form ...
5
votes
7answers
310 views

Intuitively, why does $\int_{-\infty}^{\infty}\sin(x)dx$ diverge? [duplicate]

According to Wolfram Alpha, $\int_{-\infty}^{\infty}\sin(x)dx$ does not converge. This makes no sense to me, intuitively, which I'll justify with a plot: As we see, the positive and negative areas ...
1
vote
3answers
111 views

Evaluate $\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\,dx\,dy$

I would like to compute the following, $$ \int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy $$ It is obvious that we can rewrite the integral above to, $$ ...
1
vote
1answer
109 views

Evaluate $\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$

Prove or disprove that$$\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$$ for any $a>1$. I came across with this integral evaluating inverse ...
1
vote
1answer
31 views

How to prove the non existence of this integral?

How to prove that \begin{equation}\nonumber \int_0^\infty \sin^2\left[\pi\left(x + \frac{1}{x}\right)\right]dx \end{equation} does not exist?
2
votes
4answers
78 views

Find the value of $a$ if $\int_0^\infty \frac{2x}{a}e^{\Large\frac{-x^2}{a}}\ dx=1 $

Find the value of $a$ if $$\int_0^\infty \frac{2x}{a}e^{\Large\frac{-x^2}{a}}\ dx=1 $$ I tried to use integration by parts but I didn't get a good response.
-1
votes
2answers
36 views

A simple convergent integral but not absolutely convergent.

Anybody knows a simple example for convergent function but not absolutely convergent? ( simple = easy ) Thanks for coments!!!
5
votes
3answers
424 views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
0
votes
2answers
147 views

Integral of $ \int_{-\infty}^\infty \cos (\pi t) dt$

I need to determine whether the integral $$ \int_{-\infty}^\infty cos \,(\pi t) \;dt$$ is convergent or divergent. I rewrote this improper integral as $$ \lim \limits_{x \to{-\infty}}\int_{x}^0 ...
4
votes
4answers
141 views

Question about $\frac{\sin(x)}{x}$ and $\frac{\cos(x)}{x}$

So here is my question. As known the famous integral $$ \int_0^{\infty} \frac{\sin(x)}{x}dx$$ converges an its value is $\frac{\pi}{2}$. As I was trying to solve a different integral today, after ...
0
votes
1answer
28 views

Would like to compute the limit of some integral sequence

Consider $$ \lim_{n\rightarrow \infty}\int_{\mathbb R}e^{-|x|n}e^{\frac{x^2}{2}}dx $$ The goal of an exercise I am working on is to compute the limit of the the integral above. By intuition it should ...
1
vote
2answers
60 views

Give an example of $f$ that is $C^\infty$, $\int_0^\infty f(t)dt$ converges but $f$ does not converge.

Today, my teacher asked about a real function that was integrable, but such that $\displaystyle\lim_{t\to \infty}f(t)\neq 0$ ($f$ need not have a limit). It's easy to forge an example: a function ...
0
votes
1answer
77 views

How to prove that integral of function is convergent

$\int_{0}^{\infty} \frac{(\sin(x) )}{ x} \,\mathrm dx$ and $\int_{0}^{\infty} \frac{(\sin(x) \arctan(x))}{ x} \,\mathrm dx$ These are convergent. How to prove that?? I using the comparison test. ...
4
votes
5answers
211 views

$\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$

How can I calculate the integral: $$\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx$$ ?? I got stuck.. :/ Could you give me some hint?? Do I have to use the following formula?? $\displaystyle{\sin{(A)} ...
0
votes
3answers
66 views

Evaluate integral

How do I evaluate the following integral, the answer according to Wolfram Alpha is $2$, but I keep on getting $0$ after using integration by parts.$$\frac12\int_{-\infty}^\infty x^2e^{-|x|}\ dx$$
1
vote
3answers
106 views

Evaluate logrithmic integral

Does the following expression converge? Where $n$ is positive integer $1,2,3,...$ $$\int_0^\infty(\ln x)^n dx$$
2
votes
2answers
80 views

Compute the value of the integral $\int ^{\infty}_0 \frac{1}{1+2ax+x^2} dx $

Compute the value of the integral $$\int ^{\infty}_0 \frac{1}{1+2ax+x^2} dx $$ Differentiating by $a$ didn't work for me, and I'm all out of ideas..
-1
votes
1answer
23 views

About the convergence or divergence? [closed]

Whether the following integral converge or diverge by comparison test. \begin{align*} ...
0
votes
3answers
83 views

Stuck on solving improper integral $\int_{0}^{1} \frac{\log(1+3x^2)}{x^2} dx$

Hello I have to find the convergence and solve this improper integral: $$\int_{0}^{1} \frac{\log(1+3x^2)}{x^2} dx$$ I did the convergence part.Now, for the solving I have some problems. I did the ...
0
votes
0answers
26 views

Convergence (Absolutely or conditionally) of improper integral.

I need to find out for which values of $\alpha$ the following integral converges (and find out how exactly - absolutely or contidionally): $$ \int\limits_0^{+\infty} x^\alpha \tan\left(\sin\frac ...
2
votes
2answers
121 views

How to find this improper integral? $\int_{0}^{\pi}{\frac{\sin{x}}{\sqrt{x}}dx}$

How to calculate improper integral? $$\int_{0}^{\pi}{\frac{\sin{x}}{\sqrt{x}}dx}.$$
2
votes
3answers
128 views

Value of $\int ^\infty_0 \frac{b\sin{ax} - a\sin{bx}}{x}dx$?

$$\int ^\infty_0 \frac{b\sin{ax} - a\sin{bx}}{x}dx$$ Hello guys! I'm having trouble solving this integral...looks an awful lot like an Frullani Integral, and I've tried to get it to an appropriate ...
0
votes
0answers
54 views

Find the value of the integral $\int^\infty_0 \frac{\cos(ax)}{1+x^2} dx$ [duplicate]

I have to compute the value of the integral $$ \int^\infty_0 \frac{\cos{(ax)}}{1+x^2} dx $$ It may help that $$\int_0^\infty e^{-tx}\cdot \sin(x)dx = \dfrac{1}{1+t^2}$$ but i can't find the link ...
3
votes
4answers
116 views

Compute $\int_0^\infty \dfrac{e^{-tx}\sin(x)}{x}dx$

I have to compute$$\int_0^\infty \dfrac{e^{-tx}\cdot \sin(x)}{x}dx$$ This is following a helping problem $$\int_0^\infty e^{-tx}\cdot \sin(x)dx$$ which using IPB two times turned out to be ...
3
votes
3answers
100 views

How can an improper integral have multiple values?

Integrals like this are said to dependend on the contour of integration: $$\int^{\infty}_{-\infty}\frac{x\sin x}{x^2-\sigma^2}dx=\pi e^{i\sigma}\space \mathrm{or}\quad \pi \cos\sigma $$ How is it ...
1
vote
0answers
42 views

Is this function square-integrable? Able to be Fourier expanded?

I want to do a 3-dimensional Fourier series expansion on this function$$\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left[(a+\sin (y)+\cos (z))^2+(b+\cos (x)+\sin (z))^2+(c+\sin ...
2
votes
2answers
75 views

a question about summation of series, how to prove $\int_0^\infty e^{-x}S(x)$=$\sum_{i=0}^\infty a_nn!$

If the coefficients of $\sum_{n=0}^\infty a_nx^n$ is non-negative($a_n\ge 0$ for every n),and the sum function is S(x). Also,suppose$\sum_{i=0}^\infty a_nn!$ is convergent,please prove $\int_0^\infty ...
5
votes
5answers
96 views

a question about a complex integral, I am struggling with it!

How to prove $$\int _0^1 {\ln(x)\over{1-x^2}}={-\pi^{2}\over 8}$$ My solution: If we can prove$\int _0^1 {\ln(x)\over{1-x^2}}= \lim_{n\to \infty} \int _0^1\ln(x)(1+x^2+x^4+......+x^{2n})$,then I ...
1
vote
0answers
47 views

How does one calculate $\int_0^1 \frac {\arcsin(x)}{x}dx$? [duplicate]

How can I evaluate the following? $$\int_0^1 \frac {\arcsin(x)}{x}dx.$$ Could not find a primitive, so I went for some other methods like arranging it as a double integral or introducing a ...
8
votes
2answers
259 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 ...
4
votes
3answers
96 views

Calculate the integral $\int_{0}^{+\infty }[e^{-(\frac{a}{x})^{2}} -e^{-(\frac{b}{x})^{2}}]dx$

Calculate $$\int_{0}^{+\infty }\left[e^{-(\frac{a}{x})^{2}} -e^{-(\frac{b}{x})^{2}}\right]dx,$$ with $0<a<b$ I try to construct a inner parametric integral and change the integration order, but ...
3
votes
3answers
210 views

Integral $\int_0^{\infty} \frac{x^{a-1}}{1+x} dx $ converges?

For what values ​​of $a \in \mathbb{R}$ the following integral converges? $$\int_0^{\infty} \frac{x^{a-1}}{1+x}\ dx $$ I tried to compute the integral but I stuck solving and then I tried to compare ...
1
vote
2answers
40 views

Improper integral comparison test

Having this integral $$\int_1^{\infty}\frac{3x^2+2x +1}{x^3+6x^2+x+4}$$ In order to do the comparison test at some point it gets like $$\frac{3x^2+2x +1}{x^3+6x^2+x+4}\geq \frac{1}{4x}$$ How is ...
3
votes
1answer
182 views

Integration of $\int_0^\infty \sqrt x e^{-x^2} dx$

I am having a really hard time trying to figure out what to do with this. I feel like I've tried everything but I'm obviously missing something. Any suggestions? $$\int_0^\infty \sqrt x\ e^{-x^2}\ ...
17
votes
1answer
497 views

Integral $\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}$

Please help me prove the following identity: $$\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}\quad a>0$$ This integral is from ...
8
votes
6answers
281 views

$\int_{0}^{+\infty} \frac{1-\cos t}{t} e^{-t}dt=?$

I have a question: $$\int_{0}^{\infty} \frac{1-\cos t}{t} e^{-t}dt=\ ?$$ Thanks for your any help. Thanks ahead.
2
votes
1answer
116 views

Evaluation of tricky integral

I want to evaluate the integral $$\int _ {b} ^ {\infty} \mathrm{d} x \, \frac{e ^ {x ^ {2} / s} (b^2 + 3 x ^ 2) ^ {2}}{x (x^2 + b^2)}$$, where $b$ and $s$ are positive real numbers. I thought of ...
0
votes
3answers
44 views

Convergence or Divergence using Limits

Use a comparison test to determine whether or not the following improper integrals converge or diverge $$\int_{2}^\infty \frac{1}{\ln(x)}dx.$$ I'm stuck thinking of a function to compare it to.