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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1answer
55 views

Prove that the improper integrals are equal

Prove that $$\int_0^{\infty} \frac{\cos{x}}{1+x} dx = \int_0^{\infty} \frac{\sin{x}}{(1+x)^2} dx$$ Things that I tried so far: I tried to create integral (0, infinity) cos x/1+x - sin x/(1+x)^2 ...
2
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1answer
107 views

Integral from $0$ to $\infty$ of $\frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right)$

Evaluate the integral $$ \int_0^\infty \left( \frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right) \right) dx $$ I have read about ...
6
votes
1answer
325 views

Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

We know the followings : $$\int_{0}^{\infty}\frac{{\sin}x}{x}dx=\int_{0}^{\infty}\frac{{\sin}^2x}{x^2}dx=\frac{\pi}{2},\int_{0}^{\infty}\frac{{\sin}^3x}{x^3}dx=\frac{3\pi}{8}.$$ Also, we can get ...
1
vote
1answer
20 views

question about the convergence of a function

I'll start off with what I know: I know that if I have two functions f(x) and g(x) if: $$ \int_0^\infty f(x) < \int_0^\infty g(x) $$ If g(x) converges as the values function approaches infinity, ...
33
votes
4answers
1k views

Integral $\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$$
6
votes
2answers
183 views

Compute $ I(n) = \int^{+\infty}_{-\infty} \frac{e^{n x}}{1+e^x}\, dx$

I need compute this definite integral for values 0 < n < 1. I am not sure how to begin at all. I was able to compute the indefinite integral for the special case of n = 1/2, but I am unable to ...
19
votes
1answer
317 views

A closed form for $\int_0^\infty\ln x\cdot\ln\left(1+\frac1{2\cosh x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\ln x\cdot\ln\left(1+\frac1{2\cosh x}\right)dx=\int_0^\infty\ln x\cdot\ln\left(1+\frac1{e^{-x}+e^x}\right)dx$$ I tried to ...
1
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2answers
246 views

Convergence or divergence of integral

I'm struggling with how to show that $$ \int_1^\infty \frac{x \sin x}{\sqrt{1+x^5}}dx $$ either diverges or converges. If we call the integrand $f(x)$ then $$ f(x)\leq ...
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votes
2answers
33 views

proving converge of an improper integral via riemann

I need to show if the following integral converges: $$\int_{-\infty}^{\infty}\left|\sin{1 \over x}\right|\,\mathrm dx$$ my idea for the solution is to show that the serie of rectangles that are ...
1
vote
1answer
164 views

Evaluate an Integral involving Gaussian divided by square root of a quartic polynomial

Could you please tell me, How to evaluate the integral, $\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{e^{-a(x^2+y^2)}}{\sqrt{k^2+\beta^2(x^2-y^2)^2}}dx~dy$ I already have obtained a series ...
3
votes
0answers
43 views

Convergence of a integral

The question is: exists a natural number $n \geq 2$such that $$ \displaystyle\int_{0}^{+ \infty} \displaystyle\frac{\ln r}{(1 + r^2)^{n}} r dr< \infty ?$$ I am trying to do this : i know that ...
23
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3answers
1k views

Integral $\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}\mathrm dx$

I need your assistance with evaluating the integral $$\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}dx$$ I tried manual integration by parts, but it seemed to only ...
2
votes
1answer
121 views

Prove $\frac{\sqrt\pi z^v}{2^v~\Gamma\left(v+\frac{1}{2}\right)}\int_0^\infty e^{-z\cosh t}\sinh^{2v}t~dt=\int_0^\infty e^{-z\cosh t}\cosh vt~dt$?

How to prove $\dfrac{\sqrt\pi z^v}{2^v~\Gamma\left(v+\dfrac{1}{2}\right)}\int_0^\infty e^{-z\cosh t}\sinh^{2v}t~dt=\int_0^\infty e^{-z\cosh t}\cosh vt~dt$ ? Does some formulae in ...
2
votes
2answers
102 views

For which values $a$ does the improper integral $\int_0^{\infty}\frac{\ln(1+x^2)}{x^a}dx$ converge

Find the values $a$ s.t. the integral $$\int_0^{\infty}\frac{\ln(1+x^2)}{x^a}dx$$ converges. I tried some values of $a$ by programming, it seems that for $a=2$, the integral converges, and for ...
1
vote
2answers
211 views

How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$? [duplicate]

How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$ using polar coordinates?
3
votes
4answers
189 views

Integral from zero to infinity of $\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda$

I know that the value of the integral is as follows $$\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda =z^a \frac{\Gamma(1-a)}{a}$$ However, how exactly the integral is calculated? ...
0
votes
1answer
65 views

How to integrate this integral, [closed]

$\int_{\mathbb{R}^ n} || \mathbf{x} - \mathbf{y} ||^{-k} d\mathbf{x}$? Here $k>0$.
1
vote
1answer
114 views

Improper integral $\int_0^{\infty}\frac{x^{k-1}}{1+x^n}dx$

What is the convergence condition of the following integral $$\int_0^{\infty}\frac{x^{k-1}}{1+x^n}dx$$ and how prove that if integral is convergent then ...
3
votes
3answers
231 views

Improper Integral: $\int_{-\infty}^{\infty}\frac{\log(1+x^4)}{x^4}dx$

How can I prove this? $$\int_{-\infty}^{\infty}\frac{\log(1+x^4)}{x^4}dx=\frac{2\sqrt{2}}{3}\pi$$
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1answer
71 views

How can calculate this integral?

I have the following integral as: $$\int_{0}^{\infty}x\exp\left(-\frac{x^2+a^2}{2}\right)I_0(ax)dx.$$ where $I_0(.)$ is Bessel function of zero degree. Can any one help me calculataing this integral? ...
1
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1answer
117 views

For which values of p and q does an improper integral converge

I am to find for which values of p and q the following integral converges: $$\int_0^\infty \frac{x^p}{1+x^q}\,dx\quad (q>0)$$ As I tested the limit of the above function with $\frac{x^p}{x^q}$, ...
1
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1answer
312 views

Evaluating a Real Improper Integral by Residues

I am having trouble evaluating this improper integral due to its integrand and the singularities that are present. The question reads as Show that ...
1
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1answer
35 views

Under what conditions does integrating the normal vector along a boundary make no sense?

So suppose you have an open, simply-connected, and bounded subset $D$ of $\mathbb{R}^2$ with the boundary $\partial D$. I am interested in the integral of the normal vector along the boundary, i.e., ...
0
votes
1answer
40 views

Showing Airy's Integral (Fourier Transform of e^{-ip^3/3}) Converges

Airy's integral (a.k.a. $\widehat{e^{-ip^3/3}}$ times some constant multiple) is given by $\displaystyle\text{Ai}(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}e^{-ip^3/3}dp$. Although it looks ...
2
votes
2answers
191 views

An integral from Peskin & Schroeder's QFT (2.51)

How would you solve the following integral: $$ \int_1^\infty dx \sqrt{x^2-1} \, e^{-itx}$$ where $t$ is a constant such that $t>0$?
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0answers
32 views

Limit of a singular integral

Denote $f_{\gamma}(x) =\frac { (1+\gamma)}{2} |x|^{\gamma}$. We consider: $$I(\gamma) = \int_{-1}^1\int_{-1}^1 \ln (|x-y|) f_{\gamma}(x) f_{\gamma}(y) dx dy$$ I would like to know the limit of ...
3
votes
1answer
132 views

Improper integrals are “not totally Improper”

Question is to evaluate $$\int _{-\infty}^{\infty} \frac{dx}{(x^2+a^2)^2}\text {for } a>0$$ Idea is to calculate this using complex analysis/residue theory/contour integration. Approach is ...
1
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3answers
103 views

Help with exponential integrals

I'm trying to find a nice expression for the following function \begin{equation} f_k(x)=\int_0^\infty y^k (x+y) e^{-(x+y)^2} \text{d}y. \end{equation} So far I know that \begin{equation} ...
4
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1answer
103 views

Find $\int_0^1 \mathrm{\frac{x-1}{ln(x)}}\,\mathrm{d}x$ [duplicate]

Find $\int_0^1 \mathrm{\frac{x-1}{ln(x)}}\,\mathrm{d}x$ I tryed this: $\int_0^1 \mathrm{\frac{x-1}{ln(x)}} = \int_0^1 \mathrm{\frac{x}{ln(x)}} - \int_0^1 \mathrm{\frac{1}{ln(x)}}\,\mathrm{d}x$ To ...
1
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0answers
154 views

Improper integral of odd function

I'm a student. In a recent assignment I was asked to find the mean of a Student's t multivariate distribution (which should be $\overline\mu$). I've divided the integral required to find the expected ...
1
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1answer
109 views

What is the value of $\int_{-\infty}^{\infty} \frac{e^{-ix}}{x^{2 }+ 4} dx$ [duplicate]

What is the value of: $$ \int_{-\infty}^{\infty} \frac{e^{-ix}}{x^{2}+ 4} dx $$ And also maybe the problem should be: $$ \int_{-\infty}^{\infty} \frac{e^{-3ix}}{x^{2}+ 4} dx ...
1
vote
1answer
42 views

Calculation of an integral via residue.

$$\int_{-\infty}^{\infty}{{\rm d}x \over 1 + x^{2n}}$$ How to calculate this integral? I guess I need to use residue. But I looked at its solution. But it seems too complicated to me. Thus, I asked ...
1
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2answers
173 views

Improper Integral Convergence of Positive Continuous Function

I ask for some help or hint how to deal with this question: Suppose f(x) is continuous and positive function for all $$x\ge a$$ Prove or provide a counterexample: If $$\int_{a}^\infty f(x)dx $$ ...
2
votes
2answers
86 views

Find $p$ and $q$ so that the integral converges

Find all values of $p$ and $q$ so that the below integral converges: $$ I=\int_{0}^{1} x^p \left(\log\frac{1}{x}\right)^q\;\mathrm{d}x $$ I tried and got the solution as: $q\geq0$ and ...
0
votes
1answer
75 views

Closed-form Solution of an Integral Equation

Recently, I get stuck in an equation, and I would like to obtain an closed-form solution of an equation, which is given as $$ \int_1^\infty\frac{x}{t^\alpha-x}dt=c, $$ where $\alpha>1$ and $c>0$ ...
1
vote
1answer
45 views

Integral convergent calculating

I have to calculate if this integral is convergent for any "$a$": $$ \int_2^\infty \frac{1}{x\log^a(x)} dx $$ (I made sure to not make any mistake when writing the integral from paper into this ...
3
votes
3answers
60 views

$\int_{2}^{\infty}\frac{dx}{x^2-1}$ converge or diverge

How can I find whether the integral converge or diverge. I did the following. $\int_{2}^{\infty}\frac{dx}{x^2-1}$ I did the following $\frac{a}{x-1}+\frac{b}{x+1}$ $A(x+1)+B(x-1)=1$ ...
3
votes
1answer
140 views

$\int_{-\infty}^{0}xe^{x}$ diverge or converge

How would one find whether the following improper integral converge or diverge. $\int_{-\infty}^{0}xe^{x}$ I did the following. $t\rightarrow\infty$ $\int_{t}^{0}xe^x$ I did the integration by ...
1
vote
1answer
120 views

Definite integral: $\lim\limits_{n\to\infty}\int_{-\sqrt n}^{\sqrt n}\left(1-\frac{x^2}{2n}\right)^n \,\mathrm dx$

$$\lim_{n\to\infty}\int_{-\sqrt n}^{\sqrt n}\left(1-\frac{x^2}{2n}\right)^n \,\mathrm dx $$ I tried very hard to solve this integral and it looks like while $n$ goes to infinity it will give us a ...
1
vote
6answers
270 views

if the improper integral $\int^\infty_a f(x)\,dx$ converges, then $\lim_{x→∞}f(x)=0$ [closed]

I need to prove that: $$\lim_{x→∞}f(x)=0$$ if $$\displaystyle∫^∞_af(x)\,dx$$ converges. I need a proof or an specific, and if possible simple, counterexample. Would really appreciate your help! ...
1
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3answers
1k views

Determining whether an improper integral converges or diverges.

$$\int_{1}^{\infty}\dfrac{\sqrt{x^7+2}}{x^4}\text{dx}$$ I was told to let $f(x)=\dfrac{\sqrt{x}}{x^4}$ and $g(x)=\dfrac{\sqrt{x^7+2}}{x^4}$ then find the limit as $x$ approaches $\infty$ of ...
1
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2answers
128 views

$\dfrac{\sin x}{x}$ modified improper integrals.

I am trying to evaluate this integrals: $$ \int_{-\infty}^{\infty} \! \left[\frac{\sin\left(x\right)}{x}\right]^n \, \mathrm{d}x. $$ I know how to prove it if $n=1$ using Fourier Transform, but I ...
11
votes
5answers
426 views

Proving $\int_0^1\frac{\ln(x)}{x^2-1}dx=\frac{\pi^2}{8}$

How can I prove that? $$\int_0^1\frac{\ln(x)}{x^2-1}dx=\frac{\pi^2}{8}$$ I know that ...
3
votes
2answers
145 views

Proof of my conjecture on closed form of $\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}$

Let $a$, $b\in \Bbb R^+$ and $m \in \Bbb N$ then My conjectural closed form is $$\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}\,{\rm d}x = ...
3
votes
2answers
200 views

Evaluating $\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$

I have been asked to evaluate $$\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$$. I'm deliberating on whether this indefinite integral exists or not. The integrand diverges when ...
0
votes
0answers
81 views

Limit of the integral, integral of the limit (no dominated convergence)

I'm working on the integral: $\lim_{x \rightarrow 0} \int_0^T \frac{1}{\sqrt{\pi t}} \exp \left(-\frac{x^2}{t}-t\right) \left(1-\frac{t}{T} \right) \mbox{d}t$ I'ld like to inverse the integral and ...
0
votes
4answers
400 views

improper integrals (comparison theorem)

In my assignment I have to evaluate the (improper) integral, by means of the "comparison theorem". And note whether the function is divergent or convergent. $$\int^{\infty}_{0} \frac{x}{x^3 + 1}dx$$ ...
0
votes
1answer
299 views

Improper integral of a function involving square root and absolute value.

$$\int_{-2}^{8}\dfrac{dx}{\sqrt{|2x\|}}$$ I understand that you have to split this into two integrals because at $x=0$, the function is not defined. The example showed that they split up the integral ...
5
votes
3answers
275 views

A Problem on Improper Integrals

Let $f(x)$ be continuous except at $x = 0$ and let $a > 0$. Assume that the improper integral $$\int_{0}^{a}f(x)\,dx = \lim_{\epsilon \to 0+}\int_{\epsilon}^{a}f(x)\,dx$$ exists and let $$g(x) = ...
0
votes
1answer
88 views

Convergence of improper integrals in $\mathbb{R}^n$

This is probably an elementary result, but every time I need it I'm always confused. I also looked for the solution but I could not find it, so I think this will serve well as a reference for the ...