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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2
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2answers
144 views

Evaluate the improper integral $\int_{0}^{\infty}{f(x)-f(2x)\over x}dx$, where $\lim_{x \to \infty} f(x) = L$ [duplicate]

Find $$\int_{0}^{\infty}{f(x)-f(2x)\over x}\, \mathrm{d}x$$ if $f\in C([0,\infty])$ and $\lim\limits_{x\to \infty}{f(x)=L}$. I tried denoting $\displaystyle \int{f(x)\over x}dx=F(x)$, but I don't ...
1
vote
1answer
39 views

Integral Test question

So this is the problem: http://postimg.org/image/5g815zgk5/ I am getting $\lim_{b\to\infty} 2\sec^{-1}(2b) - 2\sec^{-1}2$ Now what? What do I do with $\sec^{-1}(2b)$? What happens to a trig function ...
0
votes
2answers
58 views

Convergence of $\int_0^\infty x^\alpha \cos e^x \, dx$

I tried to solve whether this integral is convergent or not and whether that convergence is conditional or absolute for a given $\alpha$. $$\int _0^{\infty }\:\:x^{\alpha \:}\cos\left(e^x\right)\, ...
1
vote
0answers
50 views

Why does integrating a complex exponential give the delta function?

How come, when we integrate a complex exponential from $ -\infty $ to $ \infty $, we get a scaled delta function? $$ \begin{align} \int_{-\infty}^{\infty} e^{i k x} \; dk & = 2 \pi \delta \left ( ...
6
votes
1answer
66 views

closed form for $\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{dx}{(b^2+x^2)}$

closed form for : $$\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{\mathrm{dx}}{(b^2+x^2)}$$ where $\beta$ is beta function I tried with the definition of beta and i got ...
5
votes
3answers
169 views

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

How do I evaluate this integral if I suppose that $a > 0$ $$\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ \mathrm{d}x .$$ For $a=2$ I got $2\pi$ I think the result will be ...
0
votes
3answers
62 views

Use the comparison test to find whether $\int_0^\infty 1/(x^2+1)^2\,dx$ converges or not

I was thinking what function I should compare it to. If I say whether a function is smaller or bigger than this one, then I must prove that. I was thinking of (x+1)^2 but I realized that this ...
2
votes
1answer
62 views

Find the area bounded between $f(x)=\frac{\arctan(x)}{x^2}$ and $g(x)=\frac{\arctan(x)}{x^2+1}$

Find the area bounded between $$f(x)=\frac{\arctan(x)}{x^2} \quad\text{and}\quad g(x)=\frac{\arctan(x)}{x^2+1}.$$ The title says the question. The limits are from 1 to infinity. I know that I ...
2
votes
2answers
29 views

$F=w(x)=\frac{k}{x^2}$ How much work required to lift a satellite an “infinite distance” into outer space?

The satellite is 6000 lbs at earth's surface, a distance $R$ from the earth's centre (so the answer will be in terms of $R$). I know that it's supposed to be an improper integral going from 0 (?) to ...
3
votes
3answers
66 views

Strange integral test for convergence in my Analysis Script (proof flawed ?)

Today I was going through my Analysis Script which my Professor used for his course (meaning he often refers to it) and I found a Lemma called Integralcriteria for convergence of Series. I read its ...
0
votes
1answer
61 views

About odd functions and improper integrals e.g. $\int^{\infty}_{-\infty}\sin x \; dx$

Does $\displaystyle \int^{\infty}_{-\infty}\sin x \; dx$ converge? Since $\sin x$ is an odd function, and we know that in definite integrals $\displaystyle \int^{a}_{-a}\sin x \; dx=0$ then does ...
3
votes
2answers
169 views

Evaluate an Integral

Evaluate: $$\int_{0}^{\infty}\dfrac{\sin^3(x-\frac{1}{x} )^5}{x^3} dx$$ I've been stumped by this Integral and cannot think of how to evaluate it. I substituted $\dfrac{1}{x^2}=t \Rightarrow ...
1
vote
1answer
66 views

Evaluate $\int_{0}^{\frac{\pi}{4}}\frac{\sec^2 \theta }{(1-\tan \theta )}\ d \theta$

Evaluate $$\int_{0}^{\frac{\pi}{4}}\frac{\sec^2 \theta }{(1-\tan \theta )}\ d \theta$$ Here's my attempt: $$u=1-tan \theta \implies -du=\sec^2 \theta d \theta$$ Substituting back in, I get ...
0
votes
1answer
29 views

Question about negative value using the ratio convergence test for integrals

Find for what $p$, $\displaystyle \int ^{\infty}_0 x^p \arctan x dx$ converges. By parts, it's equal to: $\displaystyle \lim_{b\to \infty}\frac 1 {p+1}x^{p+1}\arctan x |^b_0- \int ^b _0\frac ...
9
votes
3answers
190 views

About the integral $\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx$

I came across the following Integral and have been completely stumped by it. $$\large\int_{0}^{1}\dfrac{\log(x)\log^2(1+x)}{x}dx$$ I'm extremely sorry, but the only thing I noticed was that the ...
5
votes
0answers
112 views

Fourier sine transform of $\frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert$

Show that $$ \int_0^{\infty} kF(k)\sin(ka)\,dk = \frac{\pi}{2}aG(a) $$ where $$ F(x) = \frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert $$ and $$ G(x) = \frac{\sin x-x\cos x}{x^4} $$ EDIT: ...
0
votes
0answers
52 views

Fourier transform of $f(x)=1/x$

I would like to compute the Fourier transform for the function: \begin{equation} f(x)=\begin{cases} 1/x&, x\in [a,b] \\ 0,& x \notin [a,b]\end{cases} \end{equation} but I cannot do the ...
3
votes
0answers
61 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
1
vote
1answer
119 views

How to integrate $\int^{\infty}_{-\infty} e^{-2\pi^2/x^2} dx$?

I am wondering how can i integrate this quantity above? Here it is again, $$\int^{\infty}_{-\infty} e^{-2\pi^2/x^2}dx.$$ Thanks a lot.
0
votes
1answer
28 views

How do I calculate values for Gamma function with complex arguments?

I can calculate the values of Gamma function for positive integer arguments using the formula $\Gamma (t) = \displaystyle\int_0^{\infty} e^{-x} x ^ {t-1} \mathrm{d}x $. Which is equal to $ (t-1)! $. ...
0
votes
1answer
21 views

Calculating improper integral limits

I am trying to calculate $\int_{-\infty }^1 {dx\over x^{1/3}}$. I have come up with $\int_{-\infty }^1 {dx\over x^(1/3)}$=$\int_{-\infty }^0 {dx\over x^{1/3}}$+$\int_{0}^1 {dx\over x^{1/3}}$. Then I ...
0
votes
1answer
48 views

wrong result for $\int_{0}^{+\infty} e^{-\sqrt{x}}dx$

I have some problem with the result of this integral: $$\int_{0}^{+\infty} e^{-\sqrt{x}}dx$$ The result should be 2 but I get ...
9
votes
4answers
608 views

Integral becomes improper after a substitution

I'm suprised about the following phenomenon which I would like to discuss with you. Consider the proper integral $$\int_{\pi/4}^{\pi/2}\frac{1}{\sin(x)}dx.$$ Since $\sin(x)$ is a diffeomorphism on ...
1
vote
1answer
11 views

Clarification about Asymptotic comparison test for Improper integrals

If I have an improper integral $\displaystyle \int_{a}^{b}f(x)dx$, and $b$ is the improper extrem, if $f(x)\sim g(x)$ for $x->b^-$, the integrals $\displaystyle \int_{a}^{b}f(x)dx$ and ...
2
votes
2answers
77 views

Help with the integral $\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2}dy$

I want to do the integral : $$I(s)=\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2} \, \mathrm{d}y$$ $s$ being a complex parameter. I tried expanding the dominator of the integrand, but this way ...
2
votes
0answers
48 views

For which polynomials $P$ the integral $\int_0^\infty x^{z-1} P(x)^{-s} dx$ is computable?

I consider the following integral: $$ I(z,s)=\int_0^\infty \frac{x^{z-1}}{(P(x))^s}dx, $$ where $P(x) = a_0 + a_1 x + \cdots + a_n x^n$ is a polynomial of degree $n \geq 2$ with $P(x) > 0$ for ...
2
votes
3answers
104 views

The asymptotic behavior of an integral

The integral in hand is $$ I(n) = \frac{1}{\pi}\int_{-1}^{1} \frac{(1+2x)^{2n}}{\sqrt{1-x^2}}\, dx $$ I dont know whether it has closed-form or not, but currently I only want to know its asymptotic ...
0
votes
3answers
78 views

Show that: $\int_{0}^{\infty}{x^2e^{-x^2}}{dx} = \frac{1}{2}\int_{0}^{\infty}{e^{-x^2}}{dx}$

I am fully uncertain of how to approach this problem: Show that: $$\int_{0}^{\infty}{x^2e^{-x^2}}{dx} = \frac{1}{2}\int_{0}^{\infty}{e^{-x^2}}{dx}$$ We've just completed the section on improper ...
0
votes
2answers
66 views

How to calculate: $\lim \limits_{x \to 0^+} x \int_{x}^{1} $ $\frac{cost}{t^{\alpha}}dt$

How can I calculate: $\lim \limits_{x \to 0^+} x \int_{x}^{1} $ $\frac{cost}{t^{\alpha}}dt$ for rach $\alpha >0$, I tried to think about this as an improper integral and substituting ...
0
votes
1answer
44 views

Convergence of an Integral series of Trig Function.

$$\int^{+\infty}_{1}\left(\tan\left(\frac{1}{x}\right)\right)^2dx$$ I just have to show in a simple comparison with a known derivative if this function converges. My guess is to substitute $1/x$ for ...
2
votes
2answers
45 views

Improper integral and residues

Evaluate $\int_0^\infty \frac{dx}{x^4+1}$ By the residue theorem $$\int_{-R}^Rf(x)dx+\int_{C_R}dz=2\pi i\sum Res(f,z_i)$$ but I have problems to evaluate it because $$z^4+1=0\Rightarrow ...
3
votes
2answers
64 views

How to find $\lim_{x\to\infty} \frac{ \int_x^1 \arctan(t^2)\, dt}{x} $ [closed]

Any tips on how to find this limit: $$\lim_{x\to\infty} \frac{ \displaystyle\int_x^1 \arctan(t^2)\, dt}{x} $$
1
vote
2answers
30 views

Convergence of Improper Integrals to find the value of Convergence

$$I=\int_{-\infty}^{+\infty}f(x)\,dx=\int_{-\infty}^{+\infty} \frac{\cos(x)}{\exp(x^2)}\,dx$$ I tried using the Limit comparison for this one: $$f(x)\leq g(x) = e^{-x^2}$$ Now I can take the ...
0
votes
3answers
68 views

Show Covergence, Integral

$$\int_{0}^{1} \frac{e^x}{x^6+x} dx$$ This how i approached the problem: 1st Step : Using partial fractions. $$e^x = A(x^5+1)+Bx^2+Cx$$ Now can i solve for $$Cx = e^x$$ and get $$C=e^{-ln(x)+x}$$ ...
0
votes
1answer
31 views

Evaluate the improper integral with residues

Evaluate $\displaystyle\int_0^\infty\frac{dx}{x^2+1}$ I have that $z_0=i$ and $z_1=-i$ are singularity points but just $z_0=i$ is in the upper plane then ...
1
vote
1answer
21 views

Improper integrals and residues

I'm already read Conway, Churchill and Marsden but I'm still with doubts when it comes to improper integrals. Where come from this relation ...
1
vote
1answer
47 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 ...
1
vote
2answers
36 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
72 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
20 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 $p\in R$: ...
3
votes
2answers
60 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
61 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
125 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
37 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
57 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
17 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
37 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.
1
vote
2answers
54 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
92 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 ...