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 - Showing convergence.

1)Show that for all $n\in\mathbb{N}$ the following is true: $\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$ for a constant $C>0$ and conclude that ...
2
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3answers
104 views

$\int_1^\infty 1/\sqrt{1+e^x}dx$

Using the comparison test I am supposed to figure out whether this integral converges or diverges. what other function should I use? Also, the inequality stating that $1/\sqrt{e^x+1}$ is larger or ...
2
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2answers
49 views

An improper integral and its convegence

I have an integral $$I(\gamma)=\int\int d^3 \mathbf{r} \, d^3 \mathbf{r}' \frac{1}{|\mathbf{r}-\mathbf{r}'|+\gamma}$$ were $\gamma$ is a positive number, $\mathbf{r},\mathbf{r}' \in \mathbb{R}^3$, ...
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3answers
40 views

Convergence of improper integral?

Consider an improper integral such that: $$I = \int_0^{+\infty} \frac{f(x)}{x}dx.$$ If $\int_0^{+\infty}f(x)dx < + \infty$, Can we conclude that the integral I converges? Thanks for any answer or ...
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2answers
41 views

Struggling with the integrability of $\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$

I know quite a lot tools to determine the integrability of functions, but in this case I really don't know where to start: $$\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$$
2
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1answer
91 views

Investigate the convergence of $\int _0^\infty \frac{\sin x^2}{x} \ dx$

Investigate the convergence of $$\int_0^\infty \frac{\sin x^2}{x} \, \mathrm{d}x$$ Is it converging? Converging absolutely? I want to use Dirichlet's test for integrals. Let $f(x) = \frac 1 x$ ...
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2answers
36 views

How can I solve this integral with the comparison theorem?

I have an integral that I am not sure how to solve with the comparison theorem to see if it is divergent or convergent. $$\int_1^\infty\frac{e^{-2x}}{\sqrt{x+16}}\;dx$$ How can I solve this with ...
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2answers
47 views

Conditional convergence and Riemann's series theorem

There are tests to determine whether an integral or sum is convergent. There are test to determine whether an integral or sum is absolutely convergent. An integral or series is said to be $\mathbf ...
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0answers
55 views

Can any one help me solve this integral ??? [on hold]

![i cannot able to solve this integral ,can any one able to solve this integral and i used integral technique but i cannot able to solve this equation the integral is with respect to x ...
2
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2answers
134 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 ...
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1answer
37 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 ...
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2answers
55 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)\, ...
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0answers
24 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
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2answers
52 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 ...
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3answers
59 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
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1answer
59 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
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2answers
27 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
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3answers
61 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 ...
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1answer
57 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
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2answers
160 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 ...
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1answer
62 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 ...
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1answer
28 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 ...
4
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3answers
128 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
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0answers
103 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: ...
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0answers
49 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 ...
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1answer
81 views

Lebesgue dominated convergence problem

This is an old exam problem: Evaluate $$\large{\lim_{n\rightarrow\infty}\int_0^{\frac{n\pi}{2}}\frac{1}{n}e^{-x}\tan{\frac{x}{n}}\,\text{d}x.}$$ My idea is to let ...
3
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0answers
58 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 ...
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1answer
100 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
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1answer
27 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)! $. ...
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1answer
20 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
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1answer
47 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
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4answers
606 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 ...
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1answer
10 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
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2answers
73 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 ...
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1answer
21 views

Improper integral: for what values of parameters exist? [closed]

$\displaystyle\int_0^\infty \dfrac{\mathrm dx}{x^p(1+x)^q}$ $\displaystyle\int_2^\infty \dfrac{e^{-px}}{\ln(x)}\mathrm dx$ Can somebody help me with those, with full problem-solve ?
2
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1answer
46 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
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3answers
99 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 ...
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3answers
77 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
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2answers
65 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
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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
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2answers
40 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
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2answers
61 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
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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
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3answers
65 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
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1answer
30 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
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1answer
18 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
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1answer
45 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
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2answers
34 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
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2answers
71 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
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0answers
18 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 $p\in R$: ...