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.

learn more… | top users | synonyms

2
votes
1answer
50 views

Convergence of the integral $\int_0^\infty \frac{\ln^3(1+x^{1/4})}{x^{1/5}+x^2}\,\arctan(x)\;dx$

How to prove that this integral is convergent? $$\int_0^\infty \frac{\ln^3(1+x^{1/4})}{x^{1/5}+x^2}\,\arctan(x)\;dx$$ I have a little experience with this kind of problem, I know we should solve the ...
3
votes
2answers
61 views

What to know about convergence of integrals

According to the values of p>0 examine the convergence of the integral: $$\int_0^{+\infty} \dfrac{\ln(1+2x^{3p})}{(x+x^2)^{4p}\arctan(x)^{1/2}}dx$$ I didn't find a good explanation about this kind of ...
1
vote
2answers
75 views

integration of $\int_{1}^{\infty } \,\left(\frac{2x^{2}+bx\text{+}a}{x(2x+a)} -1\right) \, dx=1$

i need help for this problem; Find values of a and b $$\int_{1}^{\infty} \left( \frac{2x^{2}+bx+a}{x(2x+a)} -1\right) \, dx=1$$ I very appreciate your comments and suggestions.
0
votes
0answers
35 views

Analysing the convergence of improper integral with parameter

Can you, please, check if it's right what I did: Here's the exercise: Test the convergence of the following improper integral which is defined using parameter $p\in R$: $$\int_{0}^{\infty}\frac{x^...
3
votes
2answers
72 views

Fourier transform of the 1-d Coulomb potential

Though it may sound like a physical problem, but the thing I will introduce is rather mathematical. For the Fourier transform of Coulomb potential $$ V(\vec{x})=\frac{1}{\vert x\vert} $$ I can ...
4
votes
2answers
69 views

Improper complex integration

I was trying the problem of Spiegel complex variables chapter 4 prob 93 : $$\int_{0}^{\infty}xe^{-x}\sin x\, \mathrm dx = \frac12$$ I tried with by parts and and put the limits... but the ans is not ...
2
votes
1answer
171 views

Calculation of an improper integral in the context of complex functions [duplicate]

I am facing the following improper integral: $$\int_0^\infty \frac{x^5\sin x}{(1+x^2)^3}dx.$$ Clearly the expression under the integral is a meromorphic function analytic on the nonnegative part of ...
0
votes
2answers
42 views

Convergence of $\int_{2}^{+\infty} \frac1{x \ln^\alpha x}dx$

I can't analyse the convergence of this integral: $$\int_{2}^{+\infty} \frac1{x \ln^\alpha x}dx$$ with $\alpha \in R$. I have tried to find some functions and use comparison theorem, but I haven't ...
3
votes
1answer
60 views

Improper Integral, show that $\int_0^\infty \frac{x^2}{\theta^2}f(\frac{x}{\theta})\,d\theta=x$.

Let $f$ be a function, $x>0$ and $\theta>0$, and suppose $\int_{0}^{\infty}tf(t)\,dt=1$ How could I show that $\int_0^\infty \frac{x^2}{\theta^2}f(\frac{x}{\theta}) \, d\theta =x$? I try ...
0
votes
1answer
47 views

Simplification of this fourier transform signum function

Given this equation: $$\frac{-1}{4c}[\int_{ -\infty}^{\infty}g(\varpi)Sgn(x - ct - \varpi).d\varpi -\int_{-\infty}^{\infty}g(\varpi)Sgn(x+ct - \varpi ).d\varpi ]$$ Where sgn is the signum function, ...
2
votes
1answer
79 views

Taylor expansion at discontinuous point

a) Find the Maclaurin expansion of the following function: $$f(x)=\int\limits_0^x \frac{1-e^{-t^3}}{t^2} \mathrm{d}t$$ end b) evaluate the $ \displaystyle \lim_{x \to 0^{+}} f^{(29)}\, (x) $ The ...
3
votes
1answer
55 views

Improper integral: why $\int_0^1(x^2+ x^{1/3})^{-1}\,dx$ is convergent and not $\int \frac{1}{x^2}\,dx$ ???

How do I show that $\int_0^1(x^2+ x^{1/3})^{-1}\,dx$ converges? I assume you show it on $(0,1]$. Can't seem to get my head around why this would be true.
2
votes
2answers
124 views

Convergence of improper integral with arctan

I have to analyse the convergence of $\displaystyle \int_{0}^{+\infty}\frac{1}{\arctan ^\alpha x} dx$, $\alpha \in R$ I have written: $\displaystyle \int_{0}^{c}\frac{1}{\arctan ^\alpha x} dx+\int_{c}...
3
votes
3answers
139 views

Prove that $\int_{-\infty}^{\infty} x^2 e^{-\alpha x^2} dx = \frac{1}{2} \sqrt{\frac{\pi}{\alpha^3}}$

Prove that $\displaystyle\int_{-\infty}^{\infty} x^2 e^{-\alpha x^2} dx = \frac{1}{2} \sqrt{\frac{\pi}{\alpha^3}}$. You're allowed to use the formula $\displaystyle\int_{-\infty}^{\infty} e^{-\alpha ...
-1
votes
1answer
76 views

Find a sum, terms given as ratios of improper integrals [closed]

Given $$\alpha_k =\frac{\displaystyle \int_0^\infty \frac {x^{k-1}} {e^x -1 } dx }{ \displaystyle \int_0^\infty \frac{x^{k-1}}{e^x} dx }; \quad k\in N$$ find $$\frac {\alpha_2}{1} -\frac{\alpha_6}{15} ...
0
votes
2answers
54 views

Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$

I know that the improper integral $$ \int_0^\infty\frac{1}{(1+x^2)^s}\,dx $$ is convergent for $s>0.5$ and divergent otherwise. Furthermore, it has a closed form for $s \in \mathbb{N}$ (this can ...
2
votes
0answers
118 views

Numerical integration with matrices

I have a matrix integration problem. It is based on the first integral under the section, "energy transfer efficiency and transport time" in the article, environment-assisted transport. There is a ...
2
votes
2answers
84 views

Convergence of an improper integral (with a parameter)

I'm currently trying to solve an exercise that asks for which values of $\alpha \in \mathbb{R}$ the integral $\int_0^{+\infty}f(x)\,dx$ is convergent, where $f(x)=\frac{1}{x^\alpha} \log\left(1+\...
2
votes
1answer
77 views

on the sum of an infinite series

Got stuck with this series: $$ \sum_{k=0}^\infty \frac{1}{(\theta+2+k)(\theta+1)^{k+1}} $$ which should be equal to $$ \int_0^1 \frac{t^{\theta+1}}{\theta+1-t}\textrm{d}t $$ But why? Which is the ...
2
votes
4answers
156 views

Evaluation of improper integral

I need help calculating the following improper integral: $$\int_{0}^{\infty}\frac{\cos6t-\cos4t}{t}\text{d}t$$ I tried using substitutions and expansions for the cosine function, but nothing worked. ...
0
votes
1answer
31 views

Verifying An Integral Problem

Okay, so basically I thought I got my answer fully correct, but seeing the correction, it seems I'm not. Either I'm wrong or the one who corrected the exam and sent the correction is. (It's a board ...
2
votes
2answers
21 views

Single variable improper integral

Say I have an integral of $x/(1+x^2)$ that goes from negative infinity to infinity, and then part it into two integrals $A + B$ (let $A + B = I_\text{tot}$) where $A$ and $B$ have the limits from R to ...
5
votes
3answers
140 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: $$\int_{1}^{c}\frac{\ln(1+x^p)}{\sqrt{x^2-...
0
votes
2answers
75 views

Improper integral: $\int_0^\infty \frac{sin^4x}{x^2}dx$

I have been trying to determine whether the following improper integral converges or diverges: $$\int_0^\infty \frac{sin^4x}{x^2}dx$$ I have parted it into two terms. The first term: $$\int_1^\infty \...
2
votes
0answers
35 views

Decay of reciprocal gamma function and similar functions

It is known that the reciprocal gamma function $1/\Gamma$ is entire of order 1 meaning, in particular, that for any $\varepsilon > 1$ there exists $C_\varepsilon > 0$ such that $$ \left| \frac{...
1
vote
2answers
131 views

Integral of exponential functions

On this page, there are two integrals of exponential functions. First, $$\int_{0}^{\infty} c\cdot N_{0}e^{-\lambda t}dt=c\cdot \frac{N_{0}}{\lambda}$$ How does one get this result? I got $\int_{0}^...
3
votes
3answers
182 views

how to compute this definite integral

how to compute $\displaystyle I=\int\limits_{0}^{\pi/2}\frac{x}{\tan x}\,dx$ i made $f(x)=\frac{x}{\tan x}$ and then i see that $$\begin{align} \lim_{x\to0}f(x)&=\lim_{x\to0}\frac{x}{\tan x}\\ &...
3
votes
1answer
104 views

How to evaluate such integral with pole structure?

Let's have integral: $$ I = \int \limits_{-\infty}^{\infty} \frac{e^{-\frac{x^{2}}{2}}}{x - a - i0} $$ How to evaluate it? I tried to do following: $$ \frac{1}{x -a - i0} = \int \limits_{0}^{\infty}d\...
0
votes
2answers
52 views

Evaluate $\int_{0}^{\infty}\frac{\sin^{2}\left ( t \right )}{t^{2}}dt$ with help of Laplace transform

Using the following identity $$\int_{0}^{\infty}\frac{f\left ( t \right )}{t}dt= \int_{0}^{\infty}\mathcal{L}\left \{ f\left ( t \right ) \right \}\left ( u \right )du$$ I rewrote $$\int_{0}^{\infty}\...
8
votes
2answers
210 views

Proving $\int_{0}^{\pi/2}x\sqrt{\tan{x}}\log{\sin{x}}\,\mathrm dx=-\frac{\pi\sqrt{2}}{48}(\pi^2+12\pi \log{2}+24\log^2{2}) $

When trying to solve this problem: How to Integrate $ \int^{\pi/2}_{0} x \ln(\cos x) \sqrt{\tan x}\,dx$ I found his sister integral has an interesting closed form provided my calculation is correct. ...
0
votes
1answer
38 views

Convergence and Divergence and Using Various Methods

I am totally confused with the idea of convergence and divergence and which method to use to proof it. An example is a question like this: Does this integral converge? $$\int_{12}^\infty x^{-x}\...
-1
votes
1answer
81 views

Double integration of $\frac{1}{\sqrt{x^2 + y^4}}$

I am just learning double integration. I am stuck with the following problem: $$\int_{\mathbb{R}^2}\frac{1}{\sqrt{x^2 + y^4}}\,dx\,dy$$ I am not even sure whether is integral is finite. I would ...
3
votes
1answer
111 views

Integrate $f(y)=\int_{0}^{\frac{\pi}{2}} \ln(y^2 \cos^2x+ \sin^2x) .dx$

This Integral came up while attempting another question: $$f(y)=\int_{0}^{\frac{\pi}{2}} \ln(y^2 \cos^2x+ \sin^2x) .dx$$ The suggested solution was as follows: $$f'(y) = 2y \int_{0}^{\pi/2}\...
1
vote
0answers
28 views

solving improper integral through laplace transform

My Problem is : Evaluate $\int_0^{\infty} \sin(t^2)dt $ using laplace transform: can some one give me hint to solve this.
2
votes
0answers
94 views

inequality about characteristic function

Let $X$ be a random variable with density $f(x)=|x|^{-3}1_{|x|\ge1}$ and $\phi_{X}(t)=E[e^{itX}]$. Show that $\forall t\in[-1,1] $ $$|\phi_{X}(t)-1-t^2log|t||\le3t^2$$ I noticed that $E[X]=0$, so $|\...
1
vote
1answer
41 views

Limit of an integral function

I'm stuck with this exercise: Let $f\colon (0,+\infty) \to \mathbb{R}$ such that $f(x)=\int_{x}^{x+\sin{x}}\frac{dt}{\log(1+t)}.$ Prove that $\lim _{x \to +\infty} f(x)=0.$ All I have found is ...
0
votes
1answer
51 views

Improper integral $\int_0^{1/2}\frac{\mathrm{d}t}{t^a \lvert\ln(t)\rvert^b}$

I'm working in this problem and I'm having some problems. Study the convergence of this improper integral: $$\displaystyle\int_0^{\frac12}\dfrac{\mathrm{d}t}{t^a \lvert\ln(t)\rvert^b},\quad a,...
3
votes
1answer
201 views

How does one show that $\int_0^\infty \left|\frac{\sin x} x\right| \, dx=\infty$? [duplicate]

In many place one finds accounts of how to evaluate $$ \int_0^\infty \frac{\sin x} x\,dx = \underbrace{\lim_{a\to\infty}\int_0^a}_{\text{Why view it this way?}} \frac{\sin x} x\, dx. $$ And it gets ...
3
votes
2answers
160 views

Problem with evaluating the first term of Integration by Parts

Sorry for this exceedingly silly doubt. I was trying to solve the integral $$\int_0^\infty\frac{\sin^2x}{x^2}dx$$ and I initially used Integration by Parts in this way: $$\begin{align} \int_0^\infty\...
2
votes
1answer
42 views

Finer asymptotic estimate of an integral

I'm studying the asymptotic behaviour for large $n\in \mathbb N$ of $\displaystyle \int_1^\infty \frac{1}{1+t^{n+1}}$ Using the substitution $u=(n+1)\ln(t)$, $$\displaystyle \int_1^\infty \frac{1}{1+...
1
vote
1answer
51 views

Showing existence of an improper integral by estimating the absolute value

I want to show the existence of $\int_0^{\infty}\frac{\sin(x)}{x}dx$. And my questions is: It does not help when I show that $|\int_{0}^{\infty}\frac{\sin(x)}{x}dx|<\infty$, right? Because the ...
35
votes
3answers
2k views

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

Does $$~\displaystyle{\int}_0^1\frac{\text{arctanh }x}{\tan\left(\dfrac\pi2~x\right)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$$ possess a closed form expression ? This recent post, in ...
3
votes
0answers
63 views

How to Solve this Improper Integral with six poles?

I'm trying to solve the following integral, where $a>0$, $b>0$, $y\in\mathbb{R}$ and $z\in\mathbb{R}$ are given constants: $$ \int_{-\infty}^{0} \left[ \frac{1}{(x+ia)(x-y+ib)(x-z+ib)}-\frac{1}{(...
3
votes
3answers
91 views

Calculating the integral $\int_0^1 \frac{dx}{\sqrt{-\ln(x)}}$ using Euler integrals

I'm trying to calculate the integral $\int_0^1 \frac{dx}{\sqrt{-\ln(x)}}$ using Euler integrals ($\Gamma(x)$ and $B$(x,y)$). I basically have to find a way to make that integral resemble one of the ...
0
votes
1answer
30 views

Integral equality involving partial derivatives

Update 4: I found the following, updated integral identity: $$\int_{l=-\infty}^\infty l \left. \left( \frac{\partial}{\partial x} f(x,y)\right) \right\vert_{x=y=l} \mathrm{d} l = -\int_{l=-\infty}^\...
0
votes
3answers
110 views

Find whether $\int_{1}^{\infty} \frac{sin(x)}{x} dx$ is converging or not

Find whether $\int_{1}^{\infty} \frac{sin(x)}{x} dx$ is converging or not. I tried to use comparison test or limit comparison test but could't find a suitable function. How can I determine what type ...
3
votes
1answer
102 views

For which values of $p,q$ does the integral $\int_0^1 x^p (\ln\frac{1}{x})^qdx$ converges?

For which values of $p,q$ does the integral $\int_0^1 x^p (\ln\frac{1}{x})^qdx$ converges? I use the substitution $t=1/x$ to obtain this better looking integral: $\int_1^\infty \frac{(\ln t)^q}{t^{p+...
11
votes
2answers
257 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 x}{...
1
vote
1answer
38 views

Using $g(x)=1$ for the quotient test of convergence of improper integrals

Is it ok to set $g(x)=1$ for the quotient test of convergence of improper integrals? I find it easily solves many problems, for example, show if the following converge or diverge: $\displaystyle\...
1
vote
1answer
80 views

How to get a result in Integration

$$I(a)=\displaystyle\int_{0}^{1} \frac{tan^{-1}ax}{x\sqrt{1-x^{2}}} dx$$ By using Leibniz's formula, $$I'(a)=\displaystyle\int_{0}^{1} \frac{\partial}{\partial a} \frac{tan^{-1}ax}{x\sqrt{1-x^{2}}} ...