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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0answers
34 views

whether $\int_2^{\pi/2}\log\sin x \,dx$ proper or improper

My book suggests $$\int_2^{\pi/2}\log\sin x \,dx$$ is an improper integral. But I think it is not for it is bounded on the respective interval....am I correct?
1
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1answer
15 views

Find the convolution of $x(t)*h(t)$

I am studying for an exam and have the following question: $$x(t) = u(t)$$ $$h(t) = [e^{-t}-e^{-2t}]u(t)$$ where u(t) is a unit-step function. I need to find the ...
1
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0answers
35 views

Find $p$ such that $f$ defined by $f(x)=(|x|^\frac{d}{2}+|x|^d)^{-1}$ is in $L^p(R^d)$

The function f is definited in $\Bbb R^d$ by $$f(x)=\frac{1}{(|x|^\frac{d}{2}+|x|^d)}$$ How do you find $p\in[0.+\infty]$ such that $f(x)\in L^p$ ? Any help is appreciated. My attempt: $r=|x| $, ...
2
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0answers
18 views

How to calculate this integral (with limit afterwards)?

I have to calculate this integral with limit: $$ G(m,n; E) = \lim_{\epsilon \rightarrow 0^+} \iint_{-\pi}^{+\pi} d k_x d k_y \frac{e^{i(k_x m + k_y n)}}{E+ i \epsilon + 2\cos k_x + 2\cos k_y } . $$ ...
2
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1answer
81 views

Calculate integral using beta and gamma functions

I have to calculate the following integral using beta and gamma functions: $$ \int\limits_0^1 \frac{x\,dx}{(2-x)\cdot \sqrt[3]{x^2(1-x)}} $$ I came up with this terrible solution. Firstly, let's ...
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votes
1answer
50 views

Divergence of the sum of the divergent integrals

I want to prove the divergence of: $$\int_{0}^{\infty} \frac{1}{x^2-6x+8}\mathrm dx$$ $$\int_{0}^{\infty} \frac{1}{x^2-6x+8}\mathrm dx=\int_{4}^{\infty}\frac{1}{x^2-6x+8}\mathrm dx +\int_{2}^{4} ...
4
votes
2answers
121 views

Improper integrals with singularities on the REAL AXIS (Complex Variable)

I'm having some troubles when I try to solve improper integrals exercises that have singularities on the real axis. I have made a lot of exercises where singularities are inside a semicircle in the ...
0
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1answer
31 views

If a definite integral produces a finite value, does that mean it's convergent?

$\int{_0^5\frac{x}{x-2}dx}$ This integral produces a finite value of 5+ln(9/4). However, according to Wolfram Alpha, it diverges ...
3
votes
1answer
54 views

Integral inequality of exponent

How to formally prove the following inequality - $$\int_t^{\infty} e^{-x^2/2}\,dx > e^{-t^2/2}\left(\frac{1}{t} - \frac{1}{t^3}\right)$$
2
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0answers
56 views

Can $\int\limits_0^\infty\frac{e^{-t}}{e^{c\sqrt t}-1}dt$ be evaluated in closed form?

Can the integral $$\int\limits_0^\infty\frac{e^{-t}}{e^{c\sqrt t}-1}dt$$ be evaluated in closed form using some known special functions? ($c \in \Bbb C$) This is taken from a question on ...
2
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1answer
60 views

Convergence test of the following improper integral $\int_0^\infty \frac {e^{-1/x}-1} {\ x^{2/3}}dx$

I've been trying for a couple of hours to prove the convergence of the following integral: $$\int_0^\infty \frac {e^{-1/x}-1} {\ x^{2/3}}dx$$ Eventually I understood from Wolfram-Alpha that the ...
3
votes
3answers
56 views

Does $\int_0^\infty \frac{\cos x}{1+x}$ absolutly converge?

Does the following indefinite integral converge? $\int_0^\infty \frac{\cos x}{1+x}$ converges, absolutely converges? i can say that by the Dirichlet test it does converge. i am trying to ...
3
votes
3answers
114 views

Which methods can be used to evaluate the following integral?

How can I evaluate the following integral $$ \int_{0}^{\infty} x^{-1/2} \exp({-x/2})\ dx $$ I know the answer is $\sqrt{2\pi}$.
2
votes
1answer
64 views

Integral of a Dirichlet Series

I'm stuck at a problem of an exercise list... I'd like some help to solve it :) The problem: Suppose that the Dirichlet Series $$A(s)=\lim_{N \to \infty}\sum_{n=1}^Na(n)n^{-s}$$ has abscissa of ...
1
vote
1answer
70 views

Computing a double integral $\int_{-\infty}^\infty\int_{-\infty}^\infty\frac{f(t)}{1+{(x+g(t))}^2}dt\ dx$

Let $f,g$ be continuous, with $f$ integrable. How can one evaluate $\displaystyle\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{f(t)}{1+{(x+g(t))}^2}dt\ dx$ ? Any hint would be welcome. I have ...
0
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0answers
21 views

Justify the change of order of integration in $\int^\infty_0 (\int^1_0 e^{(-ay)} sin(2bxy)dx)dy$ using a Tonelli condition

Full Question: Consider $a>0$ and $b\in R$ constants. Use a Tonelli condition to justify the change of order of integration in $\int^\infty_0 (\int^1_0 e^{(-ay)} \sin(2bxy)dx)dy$ and then prove ...
0
votes
2answers
66 views

Bounded bessel functions in an s-set projection proof

The following is an extract from Falconer's Geometry of Fractal Sets about the proof of: "...Using the definition of a Bessel function $J_0=\frac{1}{2\pi}\int^{2\pi}_0 \cos(u \cos \theta) ...
1
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1answer
27 views

Integral of a Bessel function involving rational functions

I was wondering, if there is a general solution for integrals involving Bessel functions of the form: $\int_0^\infty \frac{p(x)}{q(x)} BesselJ(0,x*r) dx $ where p(x) and q(x) are polynomals of order ...
0
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0answers
34 views

Prove $\int^\infty_0 b\sin(\frac{1}{bx})-a\sin(\frac{a}{ax}) = \ln(\frac{b}{a})$ using Frulanni integrals

Prove $\int^\infty_0 b\sin(\frac{1}{bx})-a\sin(\frac{1}{ax}) = \ln(\frac{b}{a})$ I'm supposed to use Frulanni integrals which states that $\int^\infty_0 \frac{f(bx)-f(ax)}{x}\mathrm dx$ since this ...
1
vote
2answers
48 views

Prove $\int^\infty_0 \frac{\frac{1}{1+(bx)^2}-\frac{1}{1+(ax)^2}}{x}dx = ln(\frac{a}{b})$ with Frullani Integrals

Prove $\int^\infty_0 \frac{\frac{1}{1+(bx)^2}-\frac{1}{1+(ax)^2}}{x}dx = ln(\frac{a}{b})$ I'm supposed to use Frulanni integrals and use the fact that $\int^\infty_0 \frac{f(bx)-f(ax)}{x}dx$ since ...
1
vote
0answers
26 views

How to find the compound of poisson and normal distribution?

how to find the compound distribution, if the rate of poisson distribution is normally distributed with mean and variance ? I know I have to find the integral of: $$ \frac {1} {\sigma \sqrt{2 \pi} ...
2
votes
2answers
187 views

How to compute $\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}\:\text{ for }\: a>0, b\in \mathbb{C}$?

As the title says I am trying to compute the integral $I=\displaystyle\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}$ where $a>0$ and $b$ is a complex number. For the special case of $b=-2a$, we ...
2
votes
3answers
153 views

Convergence testing of the improper integral $\int_{0}^{\infty}\frac{\ln x}{\sqrt{x}(x^2-1)}\ dx$

I've tried to test this integral for convergence for a couple of hours, actually I know that $$\int_{2}^{\infty}\frac{\ln x}{\sqrt{x}(x^2-1)}\ dx$$ converges with no problem with the help of Dirichlet ...
1
vote
1answer
64 views

How to show $\int_0^{\infty} \frac{1}{\sqrt{x}}\sin({\frac{1}{x}})dx$ converges

How to show $\int_0^{\infty} \frac{1}{\sqrt{x}}\sin({\frac{1}{x}})dx$ converges? I have that $$\frac{-1}{\sqrt{x}}\le \frac{\sin({\frac{1}{x}})}{\sqrt{x}} \le \frac{1}{\sqrt{x}}$$ but when you ...
3
votes
1answer
62 views

Need help with continuing an idea concerning showing that $4\sum\limits_{n \ge 1} a_n^2 \ge \sum\limits_{n \ge 1} \frac1{n^2}(a_1+…+a_n)^2 $

I recently encountered the following problem: If $\sum a_n^2 $ converges and $\alpha_n= \frac{a_1+...+a_n}{n}$ then show that: $$4\sum_{n \ge 1} a_n^2 \ge \sum_{n \ge 1} \alpha_n^2$$ I had an ...
1
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3answers
92 views

$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals

$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals (For reference the $\cos$ Fresnel integral is $\int^\infty_0 \cos(x^2)\, dx = \frac{\sqrt{2 \pi}}{4}$) I've tried ...
2
votes
2answers
48 views

Integral $\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$

Is this integral known to have a closed form? $$\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$$ Is there anything special about it?
4
votes
3answers
122 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
1
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2answers
48 views

Prove the relation $\frac{1}{x}$=$\int^\infty_0$ $e^{-xt}$ dt, for $x>0$. Use it to prove $\int^\infty_0$ $\frac{\sin(x)}{x}$ dx = $\frac{\pi}{2}$

Prove the relation $$\frac{1}{x} = \int^\infty_0 e^{-xt}\, \text{d}t, \text{ for } x>0.$$ Use it to prove $$\int^\infty_0\frac{\sin(x)}{x}\, \text{d}x = \frac{\pi}{2}.$$ "Hint: Use ...
4
votes
1answer
54 views

Inverse Laplace Transform of $e^{c \cdot s^2}$

I am trying to find the Inverse Laplace Transform of the function $$ F(s)=e^{c \cdot s^2} $$ where $c > 0$.
2
votes
1answer
26 views

Evaluation of an integral using nonrigorous methods

I was trying to solve the following integral $$ G(\alpha,m,n)=\int_0^{\infty}\cos(2nx)e^{-\alpha x}x^{m-1}dx;n\in N,\alpha>0,m\ge1. $$ By doing a change of variable I brought it to the integral $$ ...
3
votes
1answer
205 views

Question about computing a Complicated integral

where $\beta$ is defined like this: I'm trying to prove (2.18) but i don't know how to do, i calculated the integral but i don't find anything %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EDIT1: ...
2
votes
4answers
139 views

Convergence of $\int_0^{\infty} x \cos (x^6)\,dx$

I feel that $\int_0^{\infty} x \cos (x^6) dx$ is convergent using the regular first year definition of an integral. I have been trying to convince a university professor of this, but according to him ...
1
vote
1answer
40 views

Evaluate an integral quickly

Evaluate the integral $$\int \sqrt{x} \ln(1+x)dx $$ so we should start with the substitution: $t=\sqrt{x}$ $$ \int t\ln(1+t)dt2t = 2\int t^2\ln(1+t)dt $$ From here, it seems reasonable to ...
2
votes
1answer
72 views

Evaluating $\int_0^\infty dn \, \frac{x^n}{(3n+1)(3n+2)}$

I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate, $$\int_0^\infty dn \, ...
1
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1answer
49 views

Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
0
votes
3answers
74 views

Divergence/convergence of an integral

I am told that the following integral converges for $1<n<3$. $$ \int_{-\infty}^{+\infty} (1-e^{ix}) |x|^{-n} dx $$ I am a bit baffled. Anyone with a clue or where to start with this in order to ...
-1
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1answer
78 views

Integral $\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$

How to evaluate: $$\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$$ Basically am evaluating value of $\log(1+c\chi^2)$ where $\chi^2$ is $\chi$-squared distributed
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3answers
88 views

How $\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$ exists?

How $$\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$$ exists? It is difficult question to me. i have tried to evaluate by using fact that $$\int_{-\infty}^{\infty} f(x) \ dx =\int_{-\infty}^{0} f(x)\, dx ...
4
votes
6answers
125 views

Explain why the integral $\int_{-\infty}^\infty x \,dx$ does not exist

Why is it that $$\int_{-\infty}^\infty x \,dx$$ does not exist, but $$\lim_{N \to \infty} \int_{-N}^{N} x\,dx$$ does exist? I was thinking that it involves the fact that in the second case, the ...
1
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1answer
49 views

Improper Integral - Multiple Choice Problem - $I$

Let $f$ be a function defined $\forall~ x\geq 1$.Let $n$ denote a positive integer and let $I_n$ denote the integral $\int_1^nf(x)dx$ which is always assumed to exist. Which of the following ...
0
votes
4answers
81 views

Convergence of $\int_0 ^\infty \frac {dx}{\sqrt {1+x^3}}$

Convergence of $\int_0 ^\infty \dfrac {dx}{\sqrt {1+x^3}}$ Attempt: $\lim_{x \rightarrow \infty} \dfrac {x^{\frac{3}{2}}}{\sqrt {1+x^3}} =1$ Hence, $\dfrac {1}{x^{\frac{3}{2}}}$ and $\dfrac ...
2
votes
2answers
51 views

Improper integral calculation - limit at infinity

Will you please help me prove the following limit is zero ? $$\lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} \cos(x u) du. $$ Thanks in advance
2
votes
2answers
95 views

Unusual integral

I got a clock as a gift recently. It has a very novel face in that the hour positions are given by a complex formula. For the most part, I have been able to verify the calculations presented as ...
3
votes
5answers
87 views

How to show $\int_0^\infty\frac{dx}{x\sqrt{1-x^2}}=\pi/2$

How to show that $$\int_0^\infty\frac{dx}{x\sqrt{1-x^2}}=\frac{\pi}{2}$$ The problem is that I don't know what is $$\lim\limits_{x\to\infty}{\mathrm{arcsec}\ x}$$
0
votes
0answers
53 views

Problem about limit of an integral

I came across this question while doing some exercises on integrals, and I was wondering if I could get some help. a) Show that for $n < -1$, $\int_1^N x^n dx$ converges as $N \to\infty$, and for ...
0
votes
1answer
34 views

Differentiating an Integral

Does anyone know any general approach for something like this: $$ \frac{d}{dx}\int_{-\infty}^{x}f(x,u)du\qquad\text{or}\qquad\frac{d}{dx}\int_{x}^{\infty}f(x,u)du\qquad $$ Basically, I'm trying to ...
0
votes
0answers
26 views

Simple proof for a continuous-time linear system and impulse $\delta$?

From Schaum's Outlines of Signals & Systems: Let's work with continuous-time signals. Let $T$ be a linear time-invariant system (LTI). Input $x(t)$ can be expressed as $x(t) = ...
2
votes
2answers
61 views

Convergence of $\sum_{n=1}^{\infty} \int_n^{n+1} e^{- \sqrt x} dx$

Test the convergence of $$\sum_{n=1}^{\infty} \int_n^{n+1} e^{- \sqrt x} dx$$ Attempt: For sufficiently large $x$, we have $e^{-\sqrt x} > e ^{- x}$. I also tried solving the integral by By Parts ...
3
votes
1answer
48 views

Computing with Lebesgue integrals

This problem comes from Royden's Real Analysis, 4th ed., pg 84, #19: For a number $\alpha$, define $f(x)=x^\alpha$ for $0<x\le 1$ and $f(0)=0$. Compute $\int_0^1 f$. MY WORK: I know ...