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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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.
1
vote
2answers
52 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} ...
2
votes
3answers
88 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
74 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
49 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
62 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
36 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} ...
2
votes
1answer
72 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
101 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
29 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
19 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
108 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: ...
0
votes
2answers
40 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
9 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| ...
1
vote
2answers
42 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 ...
3
votes
3answers
160 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
73 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 ...
0
votes
2answers
37 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 ...
8
votes
2answers
143 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
23 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 ...
-1
votes
1answer
72 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 ...
0
votes
0answers
25 views

Improper integral over product of exponentials: $\int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx$

I'm looking for a way to evaluate following integral $$ \int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx $$ f(x) resembles however a complex simulation and can therefore ...
3
votes
1answer
55 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 ...
1
vote
0answers
17 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.
1
vote
0answers
80 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
40 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
47 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 ...
3
votes
1answer
115 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
147 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} ...
2
votes
1answer
36 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 ...
1
vote
1answer
39 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 ...
14
votes
1answer
206 views
+50

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

Does $~\displaystyle{\LARGE\int}_0^1\frac{\text{arctanh }x}{\tan\bigg(\dfrac\pi2~x\bigg)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$ possess a closed form expression ? This recent ...
3
votes
0answers
41 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[ ...
2
votes
3answers
80 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
26 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 = ...
0
votes
3answers
102 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
39 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 ...
11
votes
2answers
234 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 ...
1
vote
1answer
19 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: ...
1
vote
1answer
74 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} ...
1
vote
1answer
25 views

When this integral: $\int_0^{\infty}\dfrac{|\log^b(x)|}{x^a}dx$ exists

I need help to determine when integral: $\int_0^{\infty}\dfrac{|\log^b(x)|}{x^a}dx$ converges. It should be done in 2 part: $$ ...
3
votes
0answers
49 views

Double integral of symmetric polylogarithmic function over rectangular region

This question was inspired by M.N.C.E.'s wonderful response here. While exploring the possibility of generalizing his result, I found that a significant part of the problem reduced to evaluating the ...
0
votes
1answer
45 views

Determine whether a function is improperly Riemann Integrable?

My attempt: i) $$\lim_{t\to\ 0} \int_t^1 \ x^{-0.5}\ $$ = $$\lim_{t\to\ 0} 2(1^{0.5}) - 2(t^{0.5}) = 2 - 0 = 2 $$ (since $lim_{t\to\ 0^+} t^{0.5} = lim_{t\to\ 0^-} t^{0.5} = 0 $) ii) ...
5
votes
1answer
142 views

How to compute $\int_{2/\pi}^{+\infty }\ln\cos(1/x)\,dx$? [closed]

What it says in the title. If $$I=\int_{2/\pi}^{+\infty }\ln\left({\cos\left({\frac{1}{x} }\right) }\right) \, \mathrm dx,$$ how could I proceed in order to find the value of $I$?
1
vote
1answer
46 views

Integration by Parts Problem: Help in understanding why a part of it equals 0

$$4I= \int_0^{\infty} \frac{4x^3 +\sin(3x)-3\sin x}{x^5} \ \mathrm{d}x $$ $$=\frac{-1}{4} \underbrace{\left[\frac{4x^3+\sin(3x)- 3 \sin x}{x^4} \right]_0^{\infty}}_{=0} +\frac{1}{4} \int_0^{\infty} ...
2
votes
1answer
33 views

Proving an equation involving integrals and limits

I have to show the following equation: $\large\int_0^\infty \! e^{-st}\cos(\beta t) \, \mathrm{d}t=\frac{s}{s^2+\beta ^2}$ with $s>0$ I've come so far: $\large\int_0^\infty \! e^{-st}\cos(\beta ...
4
votes
3answers
107 views

Problem with Differentiation under the Integral Sign

Problem: Evaluate: $$\displaystyle\int_{0}^{\infty} \dfrac{1}{x} \left(\tan^{-1}(\pi x) - \tan^{-1}x\right)dx.$$ My incorrect attempt: $$\displaystyle\int_{0}^{\infty} \dfrac{1}{x} ...
0
votes
2answers
32 views

How to formalize my argument for existence of $\lim \limits_{\epsilon \to 0^{+}} \int_{\epsilon}^{c} \frac{\sin x}{x} dx$

I know this exists - $\frac{\sin x}{x}$ is not defined on $0$ but it has an asymptote there. However, I'm finding it difficult to formalize my argument in a way that I don't say things which might not ...
3
votes
2answers
52 views

Help with Differentiating through the Integral Sign

Problem: Evaluate $$\int_{0}^{\infty} \dfrac{\sin^3{x}}{x \cdot e^x} dx=\dfrac{A\pi}{B}-\dfrac{\tan^{-1} (C)}{D},$$ My attempt through Differentiation under the Integral Sign: Using ...
1
vote
1answer
60 views

Laplace transform of function

Assume that $f(u)=(\frac{b}{πu^3})^{1/2} e^{2b} e^{-bu} e^{-b/u}$, where $b>0.$ I am trying to calculate the Laplace transform $L\{f(u)\}(s)$ and then the $n_{th}$ derivative of this transform, ...