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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Improper Integrals Comparison Method

I have the Integral: $$\int^\infty_{20}\frac{1}{x\cdot \ln^{15}(x)}\,dx$$ I know that $$\lim_{x\to \infty}(\ln(x)) = \infty$$ Subsequently, I could substitute with $$\ln(x)$$ in the denominator and ...
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1answer
34 views

Improper Integrals Convergent or Divergent

I have the integral: $$\int_1^\infty \frac{2}{x(1+\cos^2(x^2+x+1))}dx$$ I could not figure out how to represent the equation in the denominator, so I could apply the limit. I need only to find if it ...
6
votes
4answers
505 views

Fundamental Theorem of Calculus Confusion regarding atan

According to this site, $$ \int \frac{1}{a^2 \cos^2(x) + b^2 \sin^2(x)} \,dx =\frac{1}{ab} \arctan\left(\frac{b}{a} \tan(x)\right)$$ Thus, $$ \int_0^{\pi} \frac{1}{a^2 \cos^2(x) + b^2 \sin^2(x)} \, ...
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1answer
22 views

Divergence of an integral

I am sure this is a familiar example to many, as it comes up as an example of a function which belongs to $L^p$ for $p=1$ but not $L^p$ for $p < 1$, but I am having a hard time seeing why it is ...
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1answer
39 views

Name of a particular improper integral

I am curious if there is a particular name for this, $\int\limits_{-\infty}^\infty e^{i\xi^2}d\xi$. I think it might be related the Fresnel integral but I cannot see it, any suggestions?
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3answers
212 views

Seemingly Obvious Improper Integral Property

The following seems extremely obvious, so much so that I cannot see how to prove it: If $f:[a,b]\to\mathbb{R}$ is Riemann integrable then $$\int_a^bf(x)dx=\lim_{c\to b^-}\int_a^cf(x)dx$$
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3answers
64 views

Evaluate $\displaystyle\int\limits_0^{\infty}\frac x{20}e^{-x/20} dx$

I tried doing $u$-substitution and got $-20e$ as my final answer, but I think the correct answer is just $20$. I'm not sure what I did wrong, but probably had to do with plugging in infinity... could ...
1
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1answer
39 views

Limit to find convergence of improper integral

Show the integral is convergent and find the value it converges to. $$\int_1^\infty \frac{\arctan x}{x^2}$$ I have found the indefinite integral to be $$-\frac{\arctan x}{x} + \ln|x| ...
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0answers
47 views

Why the integral converges?

Say we take one of the loveliest functions in mathematics the Gaussian which looks like this: Picture of Gaussian. By eye inspection we can say that this looks like something that could have finite ...
2
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0answers
33 views

Why do the integral and the partial sum agree for small $a$ and $m$?

Consider the following naive manipulations: \begin{align} \int_0^\infty \frac{e^{-x}}{1+ax}\:dx & = \int_0^\infty e^{-x}\frac{1}{1-(-ax)}\:dx\\ &= \int_0^\infty e^{-x} \left( ...
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2answers
31 views

Improper Integral of $(y-1)^{-3/2}$ from $0$ to $2$

improper integral $$\int_{0}^{2} \frac{1}{(y-1)^{3/2}}\, dy$$ I know it doesn't work when $y = 1$, so I split the integral, right. But then i realized, it doesn't work with $0$ either, as that would ...
1
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1answer
27 views

Laplace transform identity $F(s) = \mathcal{L}(t^{-3/2} \mathrm{e}^{-1/t})$

I'm asked to prove the following result: If $F(s)$ is the Laplace transform of $f(t) = t^{-3/2} \mathrm{e}^{-1/t}$, show that $F'(s)=-s^{-1/2}F(s)$. I'm having a lot of troubles to prove this ...
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0answers
24 views

Discussion on convergence of improper integrals

I have a general question about the convergence of improper integrals and it is this: If we have an improper integral that converges, is this similar or analogous at all to saying that a function is ...
0
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1answer
40 views

Improper convergence of this integral?

$$ \int_1^{\infty} \left\langle t\right\rangle\dfrac{\cos\left(t\right) - \sin\left(t\right)}{t^2}\,dt $$ where $\left\langle t\right\rangle$ is the rationale part of $t$. I would like to use the ...
2
votes
2answers
41 views

For what value does the following improper integral exist [duplicate]

Find $m$ such that $\displaystyle\int_{-\infty}^\infty \frac{1}{(1+x^2)^m} \, dx$ is finite. I tried to substitute $x$ with $\tan\theta$ but got stuck.
3
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1answer
80 views

Use of residues to find I=$\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$

I'm working on the problem $$I=\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$$ I found 4 singularities and i would like to use the singularities in the 1st and 2nd quadrants to solve this integral; i.e. ...
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2answers
45 views

Is this limit infinite?

I'm trying to prove that $$ \frac {\sin x}{x^{3/2}} $$ is integrable (improper sense) on $(0,1)$. I am trying to find this limit : $$ \lim_{a\to 0+} \int_a^1 \frac {\sin x}{x^{3/2}} dx $$ When I ...
3
votes
2answers
67 views

How can I show that this sequence of integrals tends to the value $\infty$?

For each $n \in \Bbb Z_+$, let $$I_n = \int_{-\infty}^\infty \frac{|\sin \left( \frac{x}{n} \right) \sin(x)|}{\frac{x}{n} x} \, dx.$$ I would like to show that $I_n \xrightarrow{n\to \infty} \infty.$ ...
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1answer
24 views

How can an improper integral approach a value?

I have $\int_1^\infty\frac {1}{x^2} dx$, which converges to 1. If the region under the curve from 1 to infinity on the function $f(x) = \frac {1}{x^2}$ has infinite area when graphed (even though the ...
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1answer
58 views

Use Cauchy's Theorem to show that if $\int_{0}^{\infty}f(x)dx$ exists, then so does $\int_{L}f(z)dz$

Suppose that $f(z)$ is analytic at every point of the closed domain $0 \leq arg z \leq \alpha$ $(0 \leq \alpha \leq 2 \pi)$, and that $\lim_{z \to \infty}z f(z) = 0$. I need to prove that if the ...
3
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1answer
25 views

How to find $\lim_{R \to \infty} \int_{0}^{R} \frac{\cos(ax)-\cos(bx)}{x} dx?$

How to find the value of $$\lim_{R \to \infty} \int_{0}^{R} \frac{\cos(ax)-\cos(bx)}{x} dx?$$
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2answers
50 views

Example of a continuous function $f$ of two variables such that $\int_I f(t,x) dt$ converges for all $x$ but does not converge uniformly.

Let $I\subset\mathbb{R}$ be an interval and $x_1,x_2\in\mathbb{R}, x_1<x_2$. I'm looking for an example of a continuous function $f:I\times[x_1,x_2]\rightarrow\mathbb{R}$ such that for all $x\in ...
2
votes
1answer
56 views

Improper integral $\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$

How do I solve this? $$\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$$ I know it's a type 3 improper integral, and I'm having issues with these. I think that I need to write it as a ...
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2answers
47 views

Finding the volume of a function rotating around the x-axis. y = xe^(-x^2)

I was working on this question for a little while and I finally got it "done". However, I am not quite sure if what I did was correct or not. Here is the details: Consider $y = xe^{-x^2}$ being ...
0
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1answer
73 views

Improper integrals in real analysis

Prove that, if $f$ is a non-negative continuous function with domain $[1,\infty$) such that the limit $$\lim_{n\to\infty}\int_{1}^{n}f(x)dx$$ (where $n$ is an integer) exists, then the improper ...
3
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3answers
319 views

Definite Integral by using properties or by indefinite integration

$$\int_{0}^{1}\left(2x\sin \frac{1}{x}-\cos \frac{1}{x}\right)dx$$ Am stuck in this question. Can't solve by applying any of the properties of definite integral. What should one do? By performing ...
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2answers
57 views

Computing the integral of the sine integral remainder function

Define $\mathrm{si}(x) := \displaystyle\int_x^\infty \frac{\sin(t)}{t}\mathrm{d}t $ for all $x>0$. I have showed, by integration by parts, that this function has convergent integral over ...
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0answers
36 views

Integral of $\sin(x)\cdot x^n$

The integral is as: $$\int_0^\infty\sin\left(x\right)\times x^{n}\;\mathbb{d}x$$ Where $n\in\mathfrak{R}$. I was able to find the convergence of this integral using Dirichlet's test, which ...
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0answers
36 views

Integral involving Bessel function $J_0$.

Is there a closed form solution to \begin{equation} \int_0^\infty \frac{J_0(kr)}{k}dk, \end{equation} where $J_0$ is Bessel function and $r$ is a constant?
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1answer
36 views

What is the relation between the gaussian integral and the volume of the n-ball?

Even if I've red other threads treating this question, it's still obscure to me what deeply relates the multiple Guassian integral $\int e^{-x^2} = \sqrt \pi$ and the area of a $n$-ball. Someone ...
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1answer
52 views

A questions about integration

Let $c,m, a$ be positive integer numbers. I have the following integral: $$2\int_{0}^{\infty}\frac{(x^2+c)^{\frac{m}{2}}}{x}\mathrm{e}^{-ax}\mathrm{d}x.$$ Although, I tried to solve it by Maple, I ...
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2answers
53 views

How to determine whether an improper integral converges or diverges?

For example: $$\int_{9}^{\infty}\frac{1}{\sqrt{x^3+1}}\,dx$$ The answer is that it converges, but why? I am so confused about this kind of questions. I tried to use the comparison test, so that ...
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1answer
26 views

Convergence with p integral

I am unsure if this integral is convergent or not based on the $p$ rule that if $P$ is greater than one, it is convergent? $$\int_\pi^\infty\frac{1+\sin(x)}{x^2} \; dx$$ and since $1+\sin(x)$ is ...
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3answers
50 views

Integrating this tricky integrand

What is a good way to solve the integral $2\int_{0}^{+\infty}x^{2}e^{-2\lambda x}.dx$ When I tried integration by parts I have some zero multiplied by infinity when evaluating the limit of ...
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0answers
56 views

Evaluation of an improper integral with complex exponential

Are there any convenient ways to calculate an integral of the form $$ \int_{-\infty}^\infty\frac{a_1 e^{j\omega\alpha}+a_2e^{j\omega\beta}}{1 + a_1a_2e^{j\omega\gamma}}d\omega$$ where ...
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1answer
83 views

Integration between infinity and -infinity

Given that $$\int_{-\infty}^{\infty}(ce^{\frac{-\alpha r}{2a}})^2dr = 1$$ express c in terms of a. I assume this is meant to be done by solving the integral, but how do you evaluate ...
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0answers
22 views

$f=\sum_1^\infty n^{-1}(-1)^n\chi_{(n,n+1]}$ is not Lebesgue integrable but improper Riemann integrable

Show that $f=\sum_1^\infty n^{-1}(-1)^n\chi_{(n,n+1]}$ is not Lebesgue integrable but $\lim_{b\to \infty} \int_0^b f(x) dx$ (the limit of the Riemann integral) exists. I'm stuck on how to show this. ...
0
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1answer
29 views

Entropy of gamma-exponential compound distribution

Following this question, I have the PDF of a gamma-exponential compound distribution as $$f(y) = \frac{\alpha\beta^{\alpha}} {(y+\beta)^{(\alpha+1)}} $$ For my application I need the entropy of ...
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2answers
56 views

Showing an improper integral converges

Im trying to evaluate $$ \int\limits_0^{\pi} \frac{ \sin^2 x}{\sqrt{x} } dx $$ This seems like a convergent integral. I know we can do it by brute force: that is: Use that $\sin^2 x = \frac{ 1 - ...
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1answer
48 views

Calculate $\int_0^\infty\frac{\sin^2t}{t^2}dt$ using Fourier transform

Calculate $\int_0^\infty\frac{\sin^2t}{t^2}dt$ using Fourier transform. Unfortunately, we have not learned Plancherel's Theorem. The only other hint that was given is ...
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2answers
57 views

How should I calculate $\int_{0}^{\frac {\pi}{2}} x\cot x \,dx$?

Is the below integral convergent? How should I calculate its value? $$\int_{0}^{\frac {\pi}{2}} x\cot x \,dx$$ Should I split the integral into two intervals?
2
votes
1answer
96 views

Finding values for integral $\iint_A \frac{dxdy}{|x|^p+|y|^q}$ converges

Given the following integral $$\iint_A \frac{dxdy}{|x|^p+|y|^q}$$ where $A=|x|+|y|>1$. How can one find for which $p$ and $q$ values the integral converges? Since the function is non-negative it ...
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0answers
51 views

How can i solve This integral special Fermi-Dirac integral at Physics?

I solved this integral $$\int _{-\infty }^{\infty }x^{2}\left( \dfrac {1} {1-e^{-x}}+\dfrac {1} {1+qe^{-x}}\right) dx=\sum _{n=1}^{\infty }\dfrac {1} {n^{3}}-\sum _{n=1}^{\infty }\dfrac {\left( ...
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1answer
37 views

Proof that the improper integral $\frac{1}{(x^2-1)}$ from $0$ to $\infty$ is divergent.

$$\int_0^\infty\frac{1}{x^2-1}$$ It was said that this integral is divergent. I have tried splitting the integral. $$\int_0^1\frac{1}{x^2-1}+\int_1^\infty\frac{1}{x^2-1}$$ Using this, I tried to ...
2
votes
1answer
25 views

Find a probability density function (pdf) $p(x)$ such that $\langle \log(x+2)\rangle$ does not exist

I would like to find the simplest example of a probability density function (pdf) $p(x)$, defined over a support $\sigma$ included in $\mathbb{R}$ (possibly infinite), such that the average $\langle ...
2
votes
1answer
186 views

upper bound formula the binomial coefficients with real valued arguments

I'm trying to prove the following. Let $n\in\mathbb{N},m\in\mathbb{N}\cup\{0\},\alpha\in (n-1,n)$ and $N\in\mathbb{N}:N\ge m+1$. Prove that \begin{align} ...
0
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0answers
64 views

Does the convergence of $\int\limits_0^\infty f(x) dx$ imply $\lim\limits_{x \to \infty } xf(x) =0$?

Let $f:[0, \infty) \to \mathbb{R}$ be a continuous function. Does the convergence of $\int\limits_0^\infty f(x) dx$ allways imply $\lim\limits_{x \to \infty } xf(x) =0$?
0
votes
1answer
36 views

Square integrable harmonic function

Let $u$ be harmonic function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} u^2 dx1...dxn < +\infty$ (1). I've got to prove that it implies $u=0$. My idea was to prove that (1) implies that $u$ ...
6
votes
3answers
141 views

Convergence of $\int_{0}^{+\infty} \frac{x}{1+e^x}\,dx$

Does this integral converge to any particular value? $$\int_{0}^{+\infty} \frac{x}{1+e^x}\,dx$$ If the answer is yes, how should I calculate its value? I tried to use convergence tests but I failed ...
0
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0answers
37 views

Integrals that don't coincide with the Riemann integral?

This is probably a lame question, but I was wondering if there exist integrals that do not coincide with Riemann's integral for function that are integrable with respect to both these integrals?