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
36 views

Find the area described by $S=\{(x,y)\mid -2<x\leq 0, 0\leq y\leq 2/\sqrt{x+2}\}$

Find the area described by $S=\{(x,y)\mid -2<x\leq 0, 0\leq y\leq 2/\sqrt{x+2}\}$. I have tried to use logic to figure out what to do but cannot figure it out. I have tried to think about this ...
0
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0answers
54 views

question regarding evaluating improper integrals with complex analysis-$\int_0^\infty\sin(x^2)\,dx$ [duplicate]

I'm asked to solve the following integral with complex analysis methods: $$\int_0^\infty\sin(x^2)\,dx$$ The thing that bothers me, is that this integral has no limit in the real plane, because the ...
1
vote
2answers
59 views

what is the integral on $[0,2]$ of $x/(3-2x)$

i know that this is an improper integral, but when you evaluate the limits as $x\to (3/2)^-$ and $x\to (3/2)^+$, you get positive and negative infinity but I am not sure if you can cancel them ...
0
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2answers
48 views

Help with an improper integral!

I need some help with an indefinite integral problem (only the $2^{\textrm{nd}}$ part thou). Problem is as follows. Consider the function $f(x) = \dfrac{\ln\!\left(x\right)}{x^{p}}$, where $p>1$ ...
3
votes
3answers
168 views

Integration of $\int_{-\infty}^{\infty} e^{-x^2 + 2x} dx$

The question is easy to phrase: Show that $$ \int_{-\infty}^{\infty} e^{-x^2 + 2x} dx $$ converges and compute its value. A first thing I did is simplifying it to $\int_{-\infty}^{\infty} e^{-y^2 + ...
1
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0answers
11 views

Doing integration by parts on exponential families

I have a problem of estimating moments of an exponential family by integration by parts. Lets consider the exponential family in its canonical form. $f(x)=e^{\theta x-\psi(\theta)}h(x)$. The ...
0
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1answer
66 views

quirk involving trig substitution?

I have reduced the following trig identity to the following which is correct. $$\int \cos^2(x)\tan^3(x)dx = \int \tan(x) - \sin(x)\cos(x)dx$$ However this next step changes the value of my ...
14
votes
4answers
470 views

How to evaluate $I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$

Prima facie, this integral seems easy to calculate,but alas, this not's case $$I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$ The numerical value is I=-1.122690024730644497584272... How to ...
5
votes
3answers
80 views

Is $\int_0^\infty x^{a-1} (1-x)^{b-1} e^{t-cx} dx$ integrable?

I am trying to evaluate the integral below. Is it even integrable? (Online integral solvers e.g. WolframAlpha could not solve the indefinite or the definite integral.) $$\int_0^\infty x^{a-1} ...
1
vote
2answers
44 views

Find the volume of this improper integral?

Find the volume of the described solid of revolution or state that it does not exist. The region bounded by $f(x)=\sqrt{\frac{(x+1)}{x^3}}$ and the $x$-axis on the interval $[1,\infty)$ is resolved ...
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2answers
53 views

Calculate improper integral using Euler's integral

I have to evaluate the following integral $$\int_0^2 \frac{dx}{\sqrt[5]{x^3(2-x)^2}}$$ Thanks in advance.
1
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1answer
46 views

Prove equality with product of improper integrals

I have to prove the following equality: $$ \int\limits_0^{+\infty} \frac{dx}{\sqrt{\cosh x}} \cdot \int\limits_0^{+\infty} \frac{dx}{\sqrt{\cosh^3 x}} = \pi. $$ The first integral Wolfram Mathematica ...
0
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0answers
55 views

Leibniz Rule in Improper Integrals?

http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf In the above link, you will find a proof and various examples of the Leibniz rule. The rule given applies to integrals with finite ...
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0answers
21 views

About the special integral forms of bessel functions

Prove that according to http://people.math.sfu.ca/~cbm/aands/page_360.htm and http://people.math.sfu.ca/~cbm/aands/page_376.htm, $Y_0(z)=\dfrac{4}{\pi^2}\int_0^\frac{\pi}{2}\cos(z\cos ...
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0answers
30 views

About the extensions from Confluent Hypergeometric Function of the Second Kind

I know that $\int_0^\infty t^{a-1}(1+t)^{c-a-1}e^{-yt}~dt=\Gamma(a)U(a,c,y)$ , where $\text{Re}(a),\text{Re}(y)>0$ . How about $\int_0^\infty t^{a-1}(1+t)^{c-a-1}(1+xt)^{-b}e^{-yt}~dt$ and ...
0
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1answer
54 views

how do i know when to reevalute limits on indefinite integral?

A) $$\int_0^{ln(3)} \frac{e^x}{e^x +2}dx$$ $$u=e^x +2$$ $$du = e^x$$ $$\int_0^{ln(x)} u^{-1}du$$ when x = ln(3), u = 5 ...
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3answers
81 views

improper integral of exponential function [duplicate]

I have a problem calculating improper integrals, this one for example, can you please help me solve it? $$\int_0^\infty t^3(e^{-t^2})dt$$ thanks in advance.
1
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1answer
66 views

Test the uniform convergence integral $\int_1^\infty\frac{ln^\alpha x}{x}\sin(x)\, dx$

Test the uniform convergence integral $$ \int_{1}^{\infty} \frac{\ln^\alpha x}{x}\, \sin x \, dx, \quad \alpha\in[1,\infty). $$ As popular tests don't work, I suspect it convergence not uniform.
1
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1answer
22 views

Integration doubt in notation.

I come across two notations while solving integrals : $1.\int_a^bf(x)dx$$2.\int_{[a,b]}f(x)dx$.For improper integrals,$1.\int_{-\infty}^\infty f(x)dx$$2.\int_{\mathbb{R}}f(x)dx$ Is there any ...
0
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2answers
49 views

Double Integral related to Gaussian Integral.

We know that $\int_{-\infty}^{\infty} e^{-x^2}dx=\sqrt{\pi}.$ Using this , how can you evaluate $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2+xy)}dxdy= ?$ Are there any standard ...
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0answers
50 views

An interesting problem of a Newsboy

This is an interesting problem of a newsboy, the problem is to identify the optimal quantity he has buy one fine day, the below equation is used for calculating the optimal quantity Q? the below ...
0
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1answer
63 views

Why is the interchange allowed?

In a solution of a book of the integral: $$\int_a^{\infty} \sum_{n=1}^{\infty} \frac{1}{(z+n)^{k+1}}\,dz, \;\; a\geq 1$$ I see the following: $$\begin{align*} ...
1
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1answer
59 views

Tricky integral with infinite limits

I've been trying for some time in solving the following infinite integral. Will residue theory be of any help here? I haven't tried that yet, but it seems no method is working effectively. I want to ...
2
votes
2answers
227 views

Calculate integral using Dirichlet integral

I have to calculate the following integral: $$ \int\limits_0^{+\infty} \left(\frac{\sin \alpha x}{x}\right)^3\,dx, $$ using the Dirichlet integral: $$ \int\limits_0^{+\infty} \frac{\sin \alpha ...
2
votes
2answers
38 views

$\int _{0}^{\infty }\! \left( {\it W} \left( -{{\rm e}^{-1 -\epsilon}} \right) +1+\epsilon \right) {{\rm e}^{-\epsilon}}{d\epsilon}={\rm e} - 1$

How to prove $\int _{0}^{\infty }\! \left( {\it W} \left( -{{\rm e}^{-1 -\epsilon}} \right) +1+\epsilon \right) {{\rm e}^{-\epsilon}}{d\epsilon}={\rm e} - 1$ where W is the Lambert W function? Maple ...
10
votes
2answers
307 views

Closer form for $\int_0^\infty\frac{(\arctan{x})^2\log^2({1+x^2})}{x^2}dx$

I Would like to know the value of this integral. $$\int_0^\infty\frac{(\arctan{x})^2\log^2({1+x^2})}{x^2}dx$$ I think ...
0
votes
2answers
51 views

Improper Integral conceptual question: show that one diverges and the other converges.

I'm asked to show that the first function is divergent, and the second is convergent. $$\int_{-\infty}^{\infty} x dx $$ $$\lim\limits_{t \to \infty} \int_{-t}^t x dx$$ I'm sure I have all the ...
2
votes
1answer
72 views

Convergence of improper integral (Dirichlet-Abel)

I need to prove that the following integral converges: $$\int_{-\infty}^{\infty}\frac{\cos(\frac{\pi}{2}x)}{(x^2+1)(x-1)}dx$$ From Dirichlet-Abel, I know that: If $\alpha:[a, +\infty]$ and $U:[a, ...
17
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1answer
446 views

$ \int_{0}^{\frac{\pi}{4}}\tan^{-1}\left(\frac{\sqrt{2}\cos3 \phi}{\left(2\cos 2 \phi+ 3\right)\sqrt{\cos 2 \phi}}\right)d\phi$

Evaluate the integral: $$\displaystyle \int_{0}^{\frac{\pi}{4}}\tan^{-1}\left(\frac{\sqrt{2}\cos3 \phi}{\left(2\cos 2 \phi+ 3\right)\sqrt{\cos 2 \phi}}\right)d\phi$$ I have no clue on how to attack ...
2
votes
3answers
78 views

Proving that $\lim\limits_{n\to\infty}\int_1^\infty\frac1{nx}dx \neq 0$

Problem: Let $f_n(x) = \frac{1}{nx}$ for $x\in[1, \infty)$. Show that $$\lim\limits_{n\to\infty}f_n(x) = 0$$ but $$\lim\limits_{n\to\infty}\int_1^\infty f_n(x)dx \neq 0.$$ My progress: The first ...
0
votes
2answers
20 views

Evaluating integral convergence

I have an the integral $$\int_{-\infty}^{6} xe^{\frac{x}{2}}\; dx$$ I know that this integral is convergent but I can not find how to evaluate its' convergence other than finding the limit of the ...
0
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1answer
44 views

Absolute Convergence (Improper integral)

I need to prove that the following improper integral converges absolutely, and I don't know how: $$\int_{-\infty}^{-4-\epsilon}\frac{\cos(x)}{(x^2+3x-4)x}dx$$ where $\epsilon$ is a small number ...
6
votes
2answers
130 views

$\int_0^{\infty} \frac{1}{x^2}\left({ \left(\sum_{n=1}^{\infty}\sin\left(\frac{x}{2^n}\right)\right)-\sin(x)}\right)\ dx$

While I was working on my stuff, another question suddenly came to mind, the one you see below $$\int_0^{\infty} \frac{ \displaystyle ...
0
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1answer
54 views

Divergence of improper integral

I need to prove that this integral diverges: $$\int_{-\infty}^{\infty}\frac{\cos{x}}{(x^2-3x-4)x}dx$$ And I don't know why. Any help? Thanks! I know that there are 3 real singularities: $x=0$, ...
3
votes
2answers
148 views

“Sequence language” translated into “function language”

Given a positive sequence $(a_n)$ which satisfies $\displaystyle \sum_{n=1}^{\infty}\frac{1}{a_n}$ converges. Let $S_n=a_1+a_2+\cdots+a_n$. Prove that $\displaystyle \sum_{n=1}^{\infty} ...
4
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3answers
142 views

Necessary condition for the convergence of an improper integral.

My calculus professor mentioned the other day that whenever we separate an improper integral into smaller integrals, the improper integral is convergent iff the two parts of the integral are ...
3
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1answer
52 views

Gaussian-like integral??

It has been a long time since I've needed to do integration... hope you can help What is the result of the following where $\alpha$ is a constant; $$\int_0^\infty ...
0
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0answers
47 views

Want Closed Form for an Integral/Summation

I am working on a mathematical model looks like this $$\sum_{t=0}^\infty {A\over e^{(k-r)t}B + e^{kt}C}$$ $k,r$ are fixed real numbers, $k-r$ is positive, $t$ is the index, and $A,B,C$ are non-zero ...
1
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1answer
84 views

Multiple self-convolution of rectangular function - integral evaluation

I am trying to find an $n$-multiple convolution of a rectangular function with itself. I have a function $f(x) = 1$ for $0<x<1$, 0 otherwise. I define $$ g_2 (y) = \int_{-\infty}^{\infty} ...
1
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2answers
71 views

Evalutation of an integral

I am trying to evaluate the integral $$ \int_{-\infty}^{\infty} \mathrm{d} w \frac{1}{w^n} \left (e^{iwx} -1 \right )^n e^{-iwx} $$ for some positive integer $n$ without much success. It doesn't ...
1
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3answers
97 views

Integral $\int_0^\infty \frac{xe^{-bx}}{\sqrt{x^2 +a^2}}dx$

Is the following integral solvable analytically? $a$ and $b$ are constants. $$ \int_0^\infty \frac{xe^{-bx}}{\sqrt{x^2 +a^2}}dx $$
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votes
1answer
93 views

Integral $ \int_{0}^\infty (\cos(4x))^{\sin(x)} \text{ dx} =\mbox{?}$ [closed]

$$ \int_{0}^\infty (\cos(4x))^{\sin(x)} \text{ dx} =\mbox{?}$$ I've tried $\frac{\pi}{2} -x = t$ (integration using substitution) but had no progress.
2
votes
2answers
103 views

How to simplify this integral?

I have managed to solve this integral by using Taylor series expansion to approximate the $e^x=1+x$. However, I am not successful due to the integral is not converged. Could you please give me a ...
2
votes
0answers
50 views

Asymptotic expansion of integral of e^(-t)/t^n

So we study $$f_{n}(x) = \int_x^{+\infty} \! \frac{e^{-t}}{t^{n}} \, \mathrm{d}t, \quad n \in \mathbb{N^{*}}$$. I've shown that for every $n$, $f_{n}(x) \sim_{+\infty} \frac{e^{-x}}{x^{n}}$. Now ...
0
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4answers
70 views

Convergence of $\int^{+\infty}_1\frac{e^{-x}}{\sin^2x}$

I'm studying the convergence of $$\int^{+\infty}_1\frac{e^{-x}}{\sin^2x}dx$$ The solution is "It is divergent" But I can't figure it out. For $+\infty$ it should converge because $\sin^2x$ is $1$ and ...
0
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0answers
40 views

convergence or absolute convergence of $\int_0^\infty \frac{\sin(ax)} {x^p}\,dx$

$$ \int_0^\infty \frac{\sin(ax)} {x^p}\,dx $$ The given integral is convergent if $0<p<2$ and absolutely convergent if $1<p<2$ How is the convergence of the integral different by the $p$ ...
0
votes
2answers
19 views

Calculate the improper integral and the Taylor series of $f(x) = \int_0^\infty {e^{-t}\over 1+x \cdot t} \,\mathrm dt$

For the given function: $$f(x) = \int_0^\infty {e^{-t}\over 1+x \cdot t} \,\mathrm dt$$ with $x>0$, calculate the Taylor series of $f(x)$ at $x=0$. I tried different stuff, but I did not get very ...
3
votes
3answers
494 views

Improper integral - equivalent definition?

Intuitively, it is rather obvious that $$\lim_{l\to\infty}\sum_{n=-\infty}^{\infty}f(n\Delta x)\Delta x = \int_{-\infty}^{\infty}f(x)dx \tag{1}$$ where $\Delta x = \frac{1}{l}$, assuming $f$ is ...
3
votes
2answers
68 views

uniform bound for sine integral function

Prove that for any $0<a<b$, $$ \left|\int_a^b\frac{\sin x}{x}\,dx\right|\le4 $$ Here is my approach. I used integration by parts to prove that LHS is bounded by $3$ when $a\ge 1$. I will be done ...
16
votes
3answers
589 views

Evaluate $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x}dx$

How evaluate this integral? $$I=\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x}\,dx$$ Note: $$\int_0^{\pi/2}\frac{x^2\log{(\sin x)}}{\sin^2x}dx=\pi\ln{2}-\frac{\pi}{2}\ln^22-\frac{\pi^3}{12}.$$ ...