Tagged Questions
4
votes
2answers
57 views
Evaluating the definite integral $\int_0^\infty \frac{e^{-kx}\sin x}x\,\mathrm dx$
How to evaluate the following integral?
$$\int_0^\infty \frac{e^{-kx}\sin x}x\,\mathrm dx$$
6
votes
4answers
128 views
How to calculate $ \int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4} $?
I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a ...
-2
votes
1answer
62 views
Is the integral $\int^{+\infty}_0 \arcsin (x)\, dx$ improper? Why not?
Is this integral $\displaystyle \int\limits^{+\infty}_0 \arcsin (x)\, dx$ improper? We have $+\infty $, but it's not improper. Why?
13
votes
1answer
95 views
Proving a formula for $\int_0^\infty \frac{\log(1+x^{4n})}{1+x^2}dx $ if $n=1,2,3,\cdots$
I came across the formula
$$\int_0^\infty \frac{\log \left(1+x^{4n} \right)}{1+x^2}dx = \pi \log \left\{2^n \prod_{k=1 ,\ k \text{ odd}}^{2n-1} \left(1+\sin \left( \frac{\pi k}{4n}\right) ...
1
vote
2answers
51 views
Determine whether this integral converges: $\int_1^\infty\frac{(x+1)\arctan x}{(2x+5)\sqrt x}$
Determine whether the next integral converges: $$\int_1^\infty\frac{(x+1)\arctan x}{(2x+5)\sqrt x}$$
I has this one on a test and lost all my points on this one. Since we were given no answers to the ...
0
votes
0answers
43 views
Evaluate $I=\int_0^\infty \mathrm{d}t\frac{1}{t}\frac{1}{t-s-\mathrm{i}\epsilon}\frac{1}{S}\log\frac{1-S}{1+S}$
Hey I wonder which from of the below integral do you guys think is easier to analyze (wrt branch points/cut poles and integration)
$$I=\int_0^\infty ...
2
votes
3answers
81 views
Is this integral improper? If yes - why?
Is this integral improper? If yes - why?
$$
\int\limits^2_0 \,\frac{1}{x-1} dx
$$
2
votes
1answer
121 views
Is there a formula for this integral
Is there a formula for the following integral?
$$I(a,b)=\int_0^1 t^{-3/2}(1-t)^{-1/2}\exp\left(-\frac{a^2}{t}-\frac{b^2}{1-t} \right)dt$$
where $a,b$ are non-zero real numbers.
7
votes
1answer
153 views
Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$
Background: Evaluation of $\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$
We can prove using the Beta-Function identity that
$$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma ...
1
vote
1answer
35 views
Proving Abel-Dirichlet's test for convergence of improper integrals using Integration by parts
I'm struggling with the following calculus question.
Let there be two functions $f,g : [a, \infty) \to \mathbb R$ such that:
$g$ is monotonic, differentiable and has a limit at zero
$f$ is ...
1
vote
0answers
38 views
how to calculate this $\int_{0}^\infty \sin(a^2x^2)\sin(b^2/x^2)dx$ [duplicate]
$$ \int_{0}^\infty \sin(a^2x^2)\sin(\frac{b^2}{x^2})dx$$
I've tried many things but none of them worked out for me.
11
votes
2answers
79 views
How can I calculate $\int_0^\infty u^3/(e^u-1) \, du$?
How can I calculate
$$\int_0^\infty \frac{u^3}{e^u-1} \, du$$
Acutally this is a part of derivation of Stefan-Boltzmann's Law.
And equation should give answer $\pi^4/15$.
0
votes
1answer
58 views
How can I evaluate $\int_0^\infty \frac{\sin x}{x} \,dx$? [may be duplicated] [duplicate]
How can I evaluate $\displaystyle\int_0^\infty \frac{\sin x}{x} \, dx$? (Let $\displaystyle \frac{\sin0}{0}=1$.)
I proved that this integral exists by Cauchy's sequence.
However I can't evaluate ...
4
votes
0answers
132 views
A difficult integral $\int_0^\infty \mathrm{d}t\frac{1}{t}\frac{1}{t-s-\mathrm{i}\epsilon}\frac{1}{X}\ln\frac{1-X}{1+X} $
Can anyone give any hints on how to rewrite this in terms of dilogarithms?
$$\int_0^\infty ...
4
votes
2answers
68 views
Problem in computing of integral by substitution.
I want to compute an integral like
$\int_0^{+\infty} \ln(1+x)e^{-x}\,\mathrm dx$.
Then denote $\mu = 1-e^{-x}$, so $x=-\ln(1-\mu)$. Substitute this into the integral, we get
$$\int_0^1 ...
9
votes
4answers
143 views
Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$
The problem is : Evaluate the integral $$\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$$
I have tried expand $\frac{1}{\cosh{ax}}$ and give the result in the following way:
First, note ...
0
votes
2answers
44 views
Not able to solve $\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $
If $p=\frac{7}{8}$ then what should be the value of $\displaystyle\int\limits_1^n \frac{g(x)}{x^{p+1}} \mathrm dx $
when $$g(x) = x \log x \quad \text{or} \quad g(x) = \frac{x}{\log x}? $$
...
1
vote
2answers
52 views
using gamma function to simplify integration
I have to evaluate $\int_0^1 x^2 \ln(\frac1x)^3 $.I used gamma function and used substitution $t=\ln (\frac {1}{x})^3$.
In this I get to integrate from $1$ to $-\infty$ with a minus sign ...
0
votes
0answers
44 views
How to solve an integral involving hyperbolic sine/cosine functions
I should solve the following tricky integral:
...
3
votes
4answers
103 views
How to compute $\int^{\infty}_{0} t^{(\frac1n-1)}\cos t \,\mathrm{d}t$?
How to calculate the below integral?
$$
\int^{\infty}_{0} \frac{\cos t}{t^{1-\frac{1}{n}}} \textrm{d}t = \frac{\pi}{2\sin(\frac{\pi}{2n})\Gamma(1-\frac{1}{n})}
$$
where $n\in \mathbb{N}$.
5
votes
2answers
161 views
How to prove a generalized integral identity
$$
\int_{0}^{\infty }\frac{t}{(e^{2\pi t}-1)(1+t^{2})}dt=-\frac{1}{4}+\frac{\gamma}{2}
$$
where $\gamma$ = Euler Gamma
$$
\int_{0}^{\infty }\frac{t}{( e^{2\pi t}-1)(1+t^{2}) ^{2}}dt=\frac{\pi^2}{24} ...
3
votes
1answer
86 views
Improper integral and special functions
I'd like to have an expression of the following integral:
$$\int_0^{+\infty} \left(\sqrt{1+x^4} - x^2\right) dx$$
in terms of some special functions (but not in the form given by Wolfram Alpha).
0
votes
2answers
156 views
Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
1
vote
1answer
51 views
Inverse Laplace Transform. Computing the integral.
This question is related to this one, but I'm hereby taking a different approach.
Problem: Solve
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their ...
5
votes
1answer
110 views
Integrate: $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
Can this integral be solved with contour integral or by some application of Residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catlan constant}$$
It has two ...
2
votes
2answers
64 views
Complex-valued Fourier integral: $ \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $
I'm working on the Fourier transform, but I don't know how to evaluate the integral:
$$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$
23
votes
2answers
256 views
$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
I need to find a closed-form for the following integral. Please give me some ideas how to approach it:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
3
votes
0answers
34 views
What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?
I want to compute the following integral:
$\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$
with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$.
...
16
votes
1answer
208 views
Integrating $\int^{\infty}_0 e^{-x^2}\,dx$ using Feynman's parametrization trick
I stumbled upon this short article on last weekend, it introduces an integral trick that exploits differentiation under the integral sign. On its last page, the author, Mr. Anonymous, left several ...
1
vote
1answer
91 views
Improper integral sin(x)/x converges absolutely, conditionaly or diverges?
$$\int_1^{\infty}\frac{\sin x}{x}dx$$
$$u=\frac{1}{x}$$
$$du=-\frac{1}{x^2}dx$$
$$dv=\sin xdx$$
$$v=-\cos x$$
$$\int_1^{\infty}\frac{\sin x}{x}dx=\frac{1}{x}(-\cos x)-\int_1^{\infty}\frac{\cos ...
8
votes
2answers
120 views
Proving that $\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}$ by induction
I need to prove
$$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$
I've seen other demonstrations of this, but they use some identities that I don't understand. ...
0
votes
1answer
36 views
integral convergence - does this integration converge
i just ran into this problem and i'm having a hard time solving it:
i would like to know if this integral converges or not, and why.
i'd prefer the normal convergence tests.
$$
...
3
votes
2answers
62 views
improper integral with square roots
Iv'e ran into this improper integral:
$$
\int_{0}^{1}(\sqrt{x}/\sqrt{1-x^6})dx
$$
I've tried to position $t=\sqrt x$ and i didn't get very close.
Any help will be greatly appreciated.
thank you,
...
2
votes
4answers
138 views
Improper Integral:$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$
I want show that this improper integral convergence: $$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$$ please help me.
3
votes
2answers
101 views
Calculating $\int_{-\infty}^\infty e^{-ax^2}e^{ibx}dx$
In my syllabus about quantum mechanics, they state that the following integral can be easily calculated:
$$\int_{-\infty}^\infty e^{-ax^2}e^{ibx}dx = \sqrt{\frac{\pi}{a}}e^{-b^2/4a}$$
if it is ...
2
votes
1answer
60 views
Calculus - improper integrals
I have a few questions from my h.w, I hope someone can help me.
the question is:
$f:[a,\infty) \rightarrow \mathbb{R}$ is a continuous and periodic function, with period of $T>0$ .
...
2
votes
2answers
72 views
Prove the next integral converges for any $x>0$: $\int_0^{\infty}t^{x-1}e^{-t}dt$
Prove the next integral converges for any $x>0$:
$$\int_0^{\infty}t^{x-1}e^{-t}dt$$
I can't find a proper way to prove that, But what i did so far was:
integration by parts:
...
1
vote
2answers
102 views
Prove the following equation: $\int_0^{\infty} \frac{\cos{(x)}}{1+x} \,\mathrm dx=\int_0^{\infty} \frac{\sin{(x)}}{(1+x)^2} \,\mathrm dx$
Prove the following equation:
$$\int_0^{\infty} \frac{\cos{(x)}}{1+x} \,\mathrm dx=\int_0^{\infty} \frac{\sin{(x)}}{(1+x)^2} \,\mathrm dx$$
0
votes
2answers
45 views
Determine if the integral converges: $\int_1^{\infty} \frac{\arctan (px)}{x^q}dx$
Determine if the integral converges:
$$\int_1^{\infty} \frac{\arctan (px)}{x^q}dx$$
where $p,q\in\Bbb R$.
2
votes
1answer
41 views
Determine if the integral converges: $\int_{-\infty}^{2} \frac{e^{3x}dx}{1+x^2}$
Determine if the integral converges:
$$\int_{-\infty}^{2} \frac{e^{3x}dx}{1+x^2}$$
I've tried this:
$$f(x)=\frac{e^{3x}}{1+x^2}>\frac{e^{\ln{3x}}}{1+x^2}=\frac{3x}{1+x^2}=g(x)$$
now since g(x) ...
3
votes
2answers
101 views
To determine whether the integral $\int_0^{\infty} \frac{\sin{(ax+b)}}{x^p} \,\mathrm dx$ converges for $p>0$
If $p >0$, determine if the following integral converges:
$$\int_1^{\infty} \frac{\sin{(ax+b)}}{x^p} \,\mathrm dx$$
What i did so far is:
If $p>1$, then $$f(x)= ...
1
vote
3answers
101 views
Solving the improper integral $\int_0^{\infty} {e^{-ax}\cos{(bx)}} dx$
$$\int_0^{\infty} {e^{-ax}\cos{(bx)}} dx$$
I know I need to use integration by part method. But I'm not sure how does the improper integral take place?
5
votes
3answers
61 views
Integration question
I have trouble in integrating the following integral.
I would appreciate any help :D
$$\int_0^1 \sqrt{-\log x}\, a\, x^{a-1}dx$$
Thanks heaps :D
The answer is $\sqrt{\pi}/2(\sqrt{a})$.
2
votes
1answer
43 views
evaluation of this integral with a fractional part
Consider the integral
$$
\int_{0}^{\infty} \left\{\frac{1}{\sqrt{t}}\right\}e^{-t/a} \mathrm dt
$$
where $ \{x\} =x-[x] $ is the fractional part of $x\in \Bbb R$. Using the representation of the ...
13
votes
2answers
466 views
Evaluating $\int_0^{\infty}\frac{e^{-x}}{1+x^2}dx$
I'm trying to evaluate $$\int_0^{\infty}\dfrac{e^{-x}}{1+x^2}dx$$
By making the substitution $x=\tan\theta$, $$\int_0^{\infty}\dfrac{e^{-x}}{1+x^2}dx=\int_0^{\frac \pi 2}\exp(-\tan\theta)d\theta$$ So ...
0
votes
2answers
118 views
Evaluate $\int\limits_0^\infty \frac{\cos(ax)}{\cos(bx)}\frac{1}{1+x^2}dx$
I would like to show that
$$\text{PV}\int_0^\infty \frac{\cos(ax)}{\cos(bx)}\frac{1}{1+x^2}dx = \frac{\pi}{2}\mathrm{sech}(b)$$
using complex analysis. $a$ and $b$ are real numbers and $a \neq b$.
...
4
votes
2answers
198 views
Integrating $\int_0^\infty \sin(1/x^2) \, \operatorname{d}\!x$
How would one compute the following improper integral:
$$\int_0^\infty \sin\left(\frac{1}{x^2}\right) \, \operatorname{d}\!x$$
without any knowledge of Fresnel equations?
I was thinking of using ...
0
votes
1answer
82 views
3 Tough improper integrals
Some weirder cases of improper integrals and probably the worse way to evaluate them.
( sorry im posting 3 even though they are pretty tough but there all very related and half my problem is how to ...
1
vote
2answers
67 views
Improper Integrals (Very Basic)
So im having a huge amount of trouble with some basic improper integrals of different types.
as they are very basic im going to post the ones i got incorrect. ( my textbook just labels converges or ...
3
votes
1answer
52 views
Convergence of this improper integral
How might I show whether $$\int_1^{\infty} \frac {x\sin x} {e^{x^3}} dx$$ converges?
Since this integrand is hard to integrate, are there standard tests (from Complex Analysis or Real Analysis) to ...
