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
24 views

How do I integrate this in terms of error function

How do I evaluate $$\dfrac{1}{\sqrt{4\pi t}}\int_0^{\infty}ye^{-\frac{(\xi-y)^2}{4t}}dy$$ in terms of $\text{erf}(x)$ ? I tried integration by parts but the integral seems to get complicated. I think ...
5
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1answer
49 views

Any general methods to calculate integral of $P(x)/Q(x)$ from $0$ to $\infty$?

In complex analysis, we have general formula for $P(x)/Q(x)$ [$P$ and $Q$ are polynomials] from minus infinity to infinity, if $ \deg Q - \deg P > 2$. Is it possible to have a general formula for ...
5
votes
1answer
55 views

How to evaluate $\int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt$?

$$ \int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt $$ I tried doing it by parts and looking for differentials but I just keep getting back to the original expression. I can't think of a clever ...
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1answer
60 views

Integral of $\sin|x|$

$$\int\sin|x|~dx$$ We have two cases: x less than zero, or x equals or higher than zero. $$\int_{-\infty}^0\sin(-x)~dx+\int_0^\infty\sin x~dx$$ Left side of this sum is equals to right side, so we ...
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1answer
27 views

Bessel's integral, how to actually evaluate?

I am just about to study Bessel functions and I have recently seen one of its integral representations given by: $$ J_ \alpha (x) = \frac{1}{\pi} \int_0 ^ \pi \cos(\alpha \tau - x\sin\tau) d\tau - ...
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0answers
27 views

convergence (absolutely) of an improper integral

$$\int_{-\infty}^\infty\frac{\sin(\sin x)}{1+\log(\lfloor|x|\rfloor! + 2)} dx$$ I need to check if this integral is absolutely convergent... I've shown it's convergent (not absolutely), according ...
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1answer
26 views

Provide examples that satisfy the following cases

Provide examples that satisfy the following cases 1) $f_n: [0, ∞)$ → R that converges uniformly to the function $f (x) = 0$ on [0, ∞) but such that $\lim_{n→∞} \int_{0}^{∞} f_n(x) dx \neq ...
1
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1answer
107 views

Difficult Improper Integral

Evaluate the improper integral $$\int_0^\infty\frac{-38x}{(2x^2+9)(3x^2+4)} dx $$ I thought about doing this through partial fractions decomposition. However, when I tried, I got some really ...
10
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1answer
87 views

Why is an equation necessarily dimensionally correct?

I have just read a fascinating proof of the value of the integral $$ \int_{-\infty}^\infty e^{-ax^2} dx, $$ which proceeds by dimensional analysis, as follows: we know that we can write $$ ...
1
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1answer
55 views

Integrating this complex function, using Residue Theorem [duplicate]

I am having a massive amount of trouble integrating this, I really have no clue how to get the answer in the book: $$\int_{-\infty}^{\infty} \frac{x^4}{1+x^8}dx$$ I know I need to find the poles ...
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2answers
29 views

Convergence of an improper integral(with parameters)

I'm trying to find solution to this problem: For what pairs (a; b) of positive real numbers does the improper integral $$ ...
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1answer
21 views

Improper integral and finding values for when they exist

please don't mark this as a replicated post, nobody is answering me on the old one. Can anyone explain what to look for next: Find the values of $p>0$ for which the following integral exists:$$I ...
1
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1answer
36 views

Finding whether improper integrals exist

For which $p>0$ does the following improper integral exist? $$\int^{\infty}_{1} x^{-p}\sin{x} \ dx$$ how do I find the value of p?
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2answers
31 views

For which $p>0$ does the following improper integral exist?

The improper integral we have is $$\int^{1}_{0} |\ln{x}|^{p}dx$$ how do I approach this? I've never done anything like this and can't find any notes on it. Thanks
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1answer
16 views

How to prove $f(x)=e^{\frac{1}{x}}$ is continous in $(0,a), a>0 $ and $\int_{0}^{a}e^{\frac{y}{x}}dx, y>0$ does not exist

I would aprecciate any advice. I'm trying to prove that in the context of a measure space, $(X,B,\lambda)$ , with $X=(0, + \infty) $, $B$ the Borel sigma-algebra and $\lambda$ the Lebesgue measure, ...
6
votes
4answers
261 views

Does the improper integral exist?

I need to find a continuous and bounded function $\mathrm{f}(x)$ such that the limit $$ \lim_{T\to\infty} \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x$$ doesn't exist. I thought about ...
0
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1answer
30 views

Show the improper integral $\int^{\infty}_0 \frac 1 y e^{-y} dy$ doesn't converge.

Show the improper integral $\int^{\infty}_0 \frac 1 y e^{-y} dy$ doesn't converge. Using Wolfram Alpha: http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605 the result ...
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0answers
17 views

Cauchy integrals over a line

Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin ...
0
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1answer
50 views

Help with finding the definite integral of $e^{\frac{2x-x^2}{2}}$?

I have this integral that I am trying to evaluate by hand, but I am encountering some difficulties. According to Wolfram Alpha, the answer seems to be: However, I do not understand how they got ...
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2answers
16 views

Does the definite integral of (1 - tanh t) from 0 to x diverge as x goes to infinity?

Decidedly in the category of things I used to know how to prove but have forgotten: Does $$ \int_0^x (1 - \tanh t) \,dt $$ converge or diverge as $x \to \infty$? (I know that the indefinite ...
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1answer
18 views

improper integral convergence and $\lim\limits_{x\to\infty}f(x)\ne 0$

Let $f:[0,\infty)\to\mathbb R$ be defined by $$f(x)=\begin{cases}1+n^2(x-n)&\text{if }n-\frac{1}{n^2}\le x\le n,\ n=2,3,...\\ 1-n^2(x-n)&\text{if }n\le x\le n+\frac{1}{n^2},\ n=2,3,...\\ ...
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0answers
25 views

Convergence of improper integral with $\lim_{x\to\infty}f(x)=\alpha$

I have a problem that says: Let $f:[a,\infty)\to\mathbb R$ be a function such that the improper integral $\int_a^{\infty}f(x)dx$ converges. Assume the existence of the finite limit ...
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0answers
16 views

Divergence of a triple integral

How to prove the following result (if ture, in the first place)? $$\nabla\cdot\int\int\int ...
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0answers
68 views

What is the solution of the integral (product of two standard normal CDFs)?

I need to compute this kind of integral: where $b>0,d>0,a,c$ and $e$ are known constants and $\Phi$ is the CDF of the standard Normal distribution.
1
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1answer
56 views

The Cantor set and integrability of $\frac{1}{x}$

Let $\chi_C$ be the characteristic function of the standard Cantor set fully contained in the interval $[0,1]$. The problem is to resolve if $\lim\limits_{\varepsilon\to 0^{+}} ...
0
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1answer
48 views

Calculus 2 Integral Question

I've been trying to resolve a calculus question and seem to be having troubles understanding exactly how to approach it. Some hints are supplied, but they don't exactly seem to help. Thanks to anyone ...
0
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1answer
51 views

Does this integral converge or diverge

Does the following integral converge or diverge? $$\int_{0}^{\infty}\left|\frac{e^{-at^2}}{\sqrt{t}}\prod_{i=1}^n\left(\frac{1}{\sqrt{t}}-r_i\right) \right|^2 dt$$ where $r_i\in\mathbb{R}$, $a>0$ ...
0
votes
2answers
44 views

Evaluating improper integral

Im trying to evaluate the improper integral $$\int_{0}^{\infty}\left( \frac{e^{i \omega t}+e^{-i \omega t}}{2}\right) e^{-st} dt$$, where $\omega$ and $s$ are real positive constants and ...
2
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3answers
69 views

Integral $\int_{-\infty}^\infty \frac{dx}{(x^2+a^2)^b}$

For Quantum Mechanics I need to normalize two wavefunctions and I haven't been able to figure out the integrals. The integrals involve a normalization constant, and 2 flexible parameters since these ...
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1answer
26 views

Show that an integral can be made as small as possible.

Consider a function $\mu(s)$ satisfying the following properties: $\mu(s) \in C^0((0,+\infty))$, $\mu(s) > 0$ and $\mu(s)$ is increasing in $s \in (0,+\infty)$, $\displaystyle \int_0^1 ...
1
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3answers
47 views

Integral of exponential integral

For real nonzero values of x, the exponential integral $\;\mbox{Ei}(x)\;$ may be defined as: $$ \mbox{Ei}(x) = \int_{-\infty}^{x} \frac{e^t}{t}\:dt $$ I have more than one reason to believe in the ...
1
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2answers
30 views

Evaluating an improper integral yields an indeterminate answer?

$$\displaystyle \int_{0.5}^1 \frac{\ln(1-t^2)}{t^2} \mathrm{d}t = \lim_{h\to 1^-} \int_{0.5}^h \frac{\ln(1-t^2)}{t^2} \mathrm{d}t =$$ $$ \lim_{h\to 1^-} \left[\ln(1-t)-\ln(1+t)-\frac{\ln(1-t^2)}{t} ...
0
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1answer
30 views

A complex-valued integral

I feel as if I'm probably just being stupid, but I'm working through Griffiths' book on quantum mechanics and I can't seem to evaluate the following integral: I(k) = $\int_{0}^{\infty}{(e^{(ik - a)x} ...
4
votes
1answer
61 views

How to compute $\int_0^{\infty}\sqrt x \exp\left(-x-\frac{1}{x}\right) \, dx$?

How to compute this integral? : $$\int_0^{\infty}\sqrt x \exp\left(-x-\frac{1}{x}\right) \, dx$$ Wolframalpha gives the answer $\dfrac{3\sqrt{\pi}}{2e^2}$, but how to compute this?
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0answers
52 views

Evaluating $\int_2^\infty \zeta(x) - 1 \,\, \mathrm{d}x$

While looking at a table of values for the zeta function, the fact that they approach $1$ made me wonder what the improper integral of the fractional part of the zeta function would be. I've found ...
0
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2answers
20 views

Show convergence of improper integral with nearest integer function

Suppose $f$ is decreasing and $\lim_{x\rightarrow\infty}f(x)=0$. Then why $$\int_0^\infty(-1)^{[x]}f(x)dx$$ converges? ($[x]$ is the nearest integer function). Any hint?
8
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1answer
163 views

Closed form for $\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi}\,\sqrt{\sin\phi+\cos\phi}}d\phi$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi}\,\sqrt{\sin\phi+\cos\phi}}d\phi$$ Its approximate numeric value is ...
2
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2answers
60 views

How to see this improper integral diverges?

$$\int^\infty_1\frac{1}{x^{1+1/x}}dx$$ I'm preparing for exams. I would also like to know what are some commonly used methods to show an improper integral diverges?
2
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4answers
93 views

Does this improper integral converge?

$$\int^\infty_0\cos x^3dx$$ I think no, because $\cos x^3$ keeps jumping between $-1$ and $1$. How to justify this rigorously?
0
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1answer
62 views

$x\rightarrow \int_{0}^{x} \frac{\operatorname{sin}(t)}{t}$ is a bounded function

I've already proved that the improper integral $\int_{0}^{\infty}\frac{\operatorname{sin}(t)}{t}$ is convergent. I don't know its limit though... I'm asked to prove that $\begin{array}{ccccc} ...
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1answer
22 views

Show convergence of improper integral

Suppose $f(x)>0$ and $f$ is continuous on $[0,\infty)$ and $$\lim_{x\rightarrow\infty}\frac{f(x+1)}{f(x)}<1$$ How to see that $\int^\infty_0f(x)dx$ converges? I think I should use definition. ...
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1answer
12 views

Evaluating an integral via differentiation under the integral sign

I recently came across Feynman Integration here, thanks to one of Lucian's links I followed from a comment thread today. It has been quite an interesting read, I must say, but sadly I am ...
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1answer
58 views

Calculate $\int_0 ^\infty\int_0 ^\infty e^{-x^2 -y^2} dx dy$

I want to calculate the following double integral: $$\int_{0}^{\infty}\int_{0}^{\infty} e^{-x^2 -y^2} \ \mathrm{d}x \ \mathrm{d}y$$ I used the change of variable $x=r\cos\theta$, $y=r\sinθ$, so I ...
1
vote
1answer
38 views

Integral from inverse Fouriertransform of 1/(1+p^2)^2

In a calculation I end up with the following integral $$\int_0 ^\infty \frac{p \sin (pr)}{(1+p^2)^2}dp , $$ could someone give me a hint how to evaluate this one? (This integral comes from the ...
0
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0answers
64 views

Different way to see that $\int_{-c}^c \text{d}{x}/x=0$?

The other day, I stumbled across the following integral: \begin{equation} \int_{-c}^c\text{d}{x}\frac{1}{x} \end{equation} with $c$, a positive real number. Now, it seems to me obvious that this ...
4
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3answers
95 views

Showing $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \log(\cos(\phi))\cos(\phi) \ d\phi = \log(4) - 2 $

This is a minor detail of a proof in 'Chaotic Billiards' by Chernov and Markarian which I foolishly decided to verify. It's page 44 of the book, during the proof that lyapunov exponents exist almost ...
1
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1answer
24 views

Definite Integral of bessel function of first kind of order one.

How to prove $\int\limits_0^\infty J_1(x)~dx=1$ ? I got $\int\limits_0^\infty J_1(x)dx=-[J_0(x)]_0^\infty$ . Please help.
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0answers
21 views

Incomplete gamma function and hypergeometric function to Meijer-G

can somebody help me to convert the incomplete gamma function and the hypergeometric function (in the forms shown below and as a function of z) into a form of Meijer-G function?
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0answers
51 views

How to calculate the improper integral $\int_{0}^{\infty}\Big(\frac{x}{\mathrm{e}^x-\mathrm{e}^{-x}}-\frac{1}{2}\Big)\frac{\mathrm{d}x}{x^2}$ [duplicate]

How to calculate the improper integral $$\int_{0}^{\infty}\Big(\frac{x}{\mathrm{e}^x-\mathrm{e}^{-x}}-\frac{1}{2}\Big)\frac{\mathrm{d}x}{x^2}$$ Do you have same idea?
1
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0answers
74 views

solution of another definite integral

Does the following integral converge or not? \begin{align} && \sum_{k=0}^{\infty} (-\varphi)^k \binom{\frac1\varphi+k}{k}\int_{-\infty}^\infty\beta x^n e^{-\beta x(k+1)}dx&& ...