Tagged Questions
1
vote
0answers
45 views
Improper integration (log, exp)
Can any one, please, help me with this problem,
$$\int_0^{\infty}\!\frac{ p (x)}{ q( x)} {(\ln(x))^{n}} e^{imx}\,dx = ?$$
where $n=1,2,\ldots,m$ and $ m \in \Bbb Z^+$, $$p (x), q(x) \in \Bbb R[x]$$
...
6
votes
3answers
73 views
Integral of a rational function: Proof of $\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}= {{\pi}\over{2}}$?
I suspect that
$$\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}=
{{\pi}\over{2}}$$
for $C>0$.
I tried $C=1$, $C=2$, $C=42$, and $C=\frac{1}{1000}$ with Wolfram ...
4
votes
1answer
60 views
Test of convergence of $\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$
I am having some trouble with this problem and don't know if I am doing it right:
$$\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$$
so the steps I have taken so far are, I split it into
...
19
votes
2answers
137 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:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
15
votes
1answer
150 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 ...
8
votes
1answer
97 views
Closed form for $\int_0^{\infty}\frac{\arctan x\ln(1+x^2)}{1+x^2}\sqrt{x}\,dx$
Please help me to find a closed form for this integral:
$$\int_0^{\infty}\frac{\arctan x\ln(1+x^2)}{1+x^2}\sqrt{x}\,dx$$
14
votes
1answer
155 views
How to prove $\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $?
I want to prove the inequality
$$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$
There are some obstacles I face: the indefinite integral cannot be expressed in terms of ...
5
votes
3answers
56 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 $\sqrt{\pi}/2(\sqrt{a})$.
3
votes
1answer
114 views
Improper integral evaluation
I'm looking for a method to evaluate the following integral:
$\displaystyle \int_0^{\infty} \left( \frac{1}{e^x - 1} - \frac{1}{x} + \frac{e^{-x}}{2} \right) \frac{1}{x} dx$
EDIT:
Using the link, ...
3
votes
0answers
115 views
Is there a closed form expression for this integral?
I've been trying to find a closed form expression/series expansion for the following integral without success:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
2
votes
3answers
61 views
Definite Integral with a discontinuty
I have the next integral:
$$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$
I have no clue how to start. At $x=0$ there is a clear discontinuity and I don't know how to solve the integral. The main ...
1
vote
1answer
164 views
Integral $\int_{-\infty}^{\infty}\frac{\cos(s \arctan(ax))}{(1+x^2)(1+a^2x^2)^{s/2}}dx$
Prove that:
$$\int_{-\infty}^{\infty}\frac{\cos(s \arctan(ax))}{(1+x^2)(1+a^2x^2)^{s/2}}dx = \frac{\pi}{(1+a)^s}$$
where $a,s \in \mathbb{R}^{+}$.
This looks difficult. What would be a good start? ...
17
votes
3answers
368 views
Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$
Many recent questions have been asked here similar to this integral
$$\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x} = 2.39587\dots$$
whose "closed form" I cannot seem to figure out. I have ...
30
votes
2answers
603 views
Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$
Prove that
$$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
8
votes
1answer
170 views
Proof of Cauchy's Beta Integral $\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}$
The Cauchy's Beta Integral is given by
$$\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}=\frac{\pi 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$
I would like to know how it is proved.
21
votes
7answers
668 views
Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$
Evaluate
$$\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$$
1
vote
3answers
203 views
Evalulate $\int_{-\infty}^{\infty}\frac{1}{(1+x^{2n})^2}dx$ by using residue theorem
I know the answer of the integral $$\int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx=\frac{\pi}{n\sin\left(\frac{\pi}{2n}\right)}$$where $n\in\mathbb{N}$.
But how to evalulate ...
1
vote
0answers
47 views
please tell me the value of the integral [duplicate]
Possible Duplicate:
What is the value of $\displaystyle\lim_{R\to\infty} {\int_R^{\infty}{r^ne^{-r^2/2}dr}\over{R^{n-1}e^{-R^2/2}}}$
please tell me the value of the integral:
...
0
votes
0answers
229 views
1
vote
0answers
89 views
Closed-form expression of the following double integral
How can I find closed-form expression of the following double integral
$$\int_0^{\frac{\pi}4} \int_0^\infty \frac{dr \, d\phi}{ u_1^2 + u_2^2 r + 2 u_1 u_2 \sqrt{r} \cos \phi}?$$ Please help me as ...
6
votes
1answer
169 views
Show $\int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t),$ for $t \gt 0$
The problem is to show
$$\int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t),$$
for $t \gt 0$.
I'm pretty stuck. I thought about integration by parts and couldn't get anywhere with the integrand in ...
0
votes
1answer
145 views
Evaluating $ \int_0^{\infty } \exp\left(-g x-\frac{x^2}{2}-\frac{x^2 z}{1-z}\right) x^k \sin(hx) \, dx $
I'm attempting to evaluate the following integral, so far, with little success. Any help would be appreciated:
$$
\ \int_0^{\infty } \exp\left(-g x-\frac{x^2}{2}-\frac{x^2 z}{1-z}\right) x^k \sin(hx) ...
8
votes
2answers
223 views
Evaluating $ \int_1^{\infty} \frac{\{t\} (\{t\} - 1)}{t^2} dt$
I am interested in a proof of the following.
$$ \int_1^{\infty} \dfrac{\{t\} (\{t\} - 1)}{t^2} dt = \log \left(\dfrac{2 \pi}{e^2}\right)$$
where $\{t\}$ is the fractional part of $t$.
I obtained a ...
4
votes
2answers
376 views
Definition of the gamma function
I know that the Gamma function with argument $(-\frac{1}{ 2})$ -- in other words $\Gamma(-\frac{1}{2})$ is equal to $-2\pi^{1/2}$. However, the definition of $\Gamma(k)=\int_0^\infty t^{k-1}e^{-t}dt$ ...
1
vote
2answers
92 views
Integration to prove indeterminate
I am doing a problem with PDF's and this integral is a roadblock.
I have $f(x)=\dfrac{200}{(x+10)^3}$ where $x>0$ and have solved that $\mathbb{E}(X)=10$.
I am working on finding ...
1
vote
0answers
140 views
Integrating the exponential and the logarithmic function together.
Ok, I want to find
$$\int\limits_0^t {{e^u}\log udu} $$ and $$\int\limits_0^t {{e^{ - u}}\log udu} $$
I'm thinking as follows
$$d\left( {{e^u}\log u} \right) = {e^u}\log udu + ...
5
votes
2answers
276 views
Does $\int_{0}^{\infty} \frac{dx}{\sqrt{x^3+x}}$ converge?
I'd like your help with checking whether $\int_{0}^{\infty} \frac{dx}{\sqrt{x^3+x}}$ converges or not. Here are the steps which led me to conclude that the integral does converge, but I'm not really ...
3
votes
2answers
336 views
Limit of the integral is the square root of pi over 2
$$\bar L = \displaystyle \limsup_{n\to\infty} \frac{1}{\sqrt{n}} \int_0^\infty e^{-x}\left(1+\frac{x}{n}\right)^n ~dx$$
How do you show the limit superior is finite?
I actually am relatively certain ...
2
votes
2answers
251 views
Trouble with integrating $\frac{\arctan (x)}{x}$
I have a function $F(x)$ that is defined as $\int_0^x f(t) dt$ which I'm trying to find the limit of when $x$ approaches infinity. Previously in the assignment, the function $f(x)$ was defined as ...
