All Questions
99
questions
2
votes
0
answers
247
views
Is it possible to evaluate $\int_{0}^{\frac{\pi}{2}} e^{-(\pi \tan(x) - 1)^2} \, dx$?
How to evaluate $$\int_{0}^{\frac{\pi}{2}} e^{-(\pi \tan(x) - 1)^2} \, dx$$
Source: I created this integral so I don’t know the closed form
I tried Wolfram Alpha, but Wolfram Alpha is unable to ...
0
votes
1
answer
120
views
How to integrate $\int_{2}^{\infty} \frac{\pi(x) \ln(x^{\sqrt{x}}) \cdot (x^2 + 1)}{(x^2 - 1)^3} \,dx$
How to integrate $$\int_{2}^{\infty} \frac{\pi(x) \ln^2(x^{\sqrt{x}}) \cdot (x^2 + 1)}{(x^2 - 1)^3} \,dx \quad?$$
Wolfram gives the numerical value
$$\int_{2}^{\infty} \frac{\pi x (1 + x^2) \log^2(x^{\...
3
votes
1
answer
183
views
Improper Integral $\int_{0}^{\infty} \log(t) t^{-\frac{1}{2}} \exp\left\{-t\right\} dt$
Background
Hi. I am currently writing my undergraduate thesis which mainly revolves around the generalized log-Moyal distribution pioneered by Bhati and Ravi (see here). In the aforementioned article, ...
- 399
7
votes
2
answers
285
views
Show that $\int_{-\infty}^\infty \frac{e^x}{e^{2x}+e^{2a}}\frac{1}{x^2+\pi^2}dx = \frac{2\pi e^{-a}}{4a^2+\pi^2}-\frac{1}{1+e^{2a}}$
Show that\begin{align*}
\int_{-\infty}^\infty \frac{e^x}{e^{2x}+e^{2a}}\frac{1}{x^2+\pi^2}dx = \frac{2\pi e^{-a}}{4a^2+\pi^2}-\frac{1}{1+e^{2a}}
\end{align*}where $a\in \mathbb{R}$.
My SOLUTION
Let $\...
4
votes
1
answer
140
views
How to evaluate $\int_0^{\infty } \left(\frac{1}{(x+1)^2 \log (x+1)}-\frac{\log (x+1) \tan ^{-1}(x)}{x^3}\right) \, dx$
How to evaluate $$\int_0^{\infty } \left(\frac{1}{(x+1)^2 \log (x+1)}-\frac{\log (x+1) \tan ^{-1}(x)}{x^3}\right) \, dx = G - \gamma + \frac{1}{4} \pi \log 2 - \frac{3}{2}.$$
I made some progress.
...
14
votes
4
answers
665
views
How to evaluate $\int_{0}^{\frac{\pi}{2}} \frac{\cos(x)}{(1 + \sqrt{\sin(2x)})^n} \,dx$
How to evaluate $$\int_{0}^{\frac{\pi}{2}} \frac{\cos(x)}{(1 + \sqrt{\sin(2x)})^n} \,dx$$
My attempt
The transformation of $x \rightarrow \frac{\pi}{2}-x$ yields
$$ \int_{0}^{\frac{\pi}{2}} \frac{\cos(...
6
votes
2
answers
165
views
Integrating $\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx$
how to integrate $$\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx$$
Attempt
$$=\int_{0}^{1} \left(\frac{\arctan(x) - x}{x^2}\right)^2 \,dx = \int_{0}^{1} \frac{1}{x^4} \cdot (\arctan(x) -...
5
votes
1
answer
125
views
A challenging Integral Involving Logarithmic and Trigonometric Functions
Question: How to evaluate $$\frac{1}{2\sqrt{2}} \int_{0}^{\frac{\pi}{2}} \frac{\log(1 + \tan y)}{(\cos y + \sqrt{2} \sin(y + \frac{\pi}{4})) \sqrt{1 + \sqrt{2} \sin(2y + \frac{\pi}{4})}} \, dy = G$$
...
4
votes
2
answers
221
views
Calculate $\int_0^\infty \frac{\sin\left(x^2 - \arctan\left(\frac{1}{x^2}\right)\right)}{\sqrt{1 + x^4}} \, dx $
Calculate $$\int_0^\infty \frac{\sin\left(x^2 - \arctan\left(\frac{1}{x^2}\right)\right)}{\sqrt{1 + x^4}} \, dx =? $$
$$\sin(\arctan(x^2)) = \frac{x^2}{\sqrt{1 + x^4}}; \quad \cos(\arctan(x^2)) = \...
user1254447
0
votes
1
answer
50
views
How to approach this integral? $ \int_0^\Theta {\frac{1}{a+b\cot\left({\pi{x}^c}\right)}} dx $
Could someone suggest how to approach this integral:
$$
\int_0^\Theta {\frac{1}{a+b\cot\left({\pi{x}^c}\right)}} dx
$$
where $a$, $b$, $c$ are scaler values, and $0<\Theta<1$. I have tried some ...
1
vote
0
answers
231
views
Improper Integral Involving Hyperbolic Cotangent
I am trying to evaluate the following integral:
$$\int_0^\infty \frac{\frac{1}{x}-\pi\coth(\pi x)}{x^2+4}dx$$
I'm not sure if a closed-form exists, so far I only know the decimal approximation to be $\...
- 63
6
votes
2
answers
238
views
Evaluating $\int_{0}^{\infty} \left( \text{coth} (x) - x \text{csch}^2 (x) \right) \left( \ln \left( \frac{4 \pi^2}{x^2} + 1 \right) \right) \, dx$
How can the following improper integral be evaluated?
$$\int_{0}^{\infty} \left( \text{coth} (x) - x \text{csch}^2 (x) \right) \left( \ln \left( \frac{4 \pi^2}{x^2} + 1 \right) \right) \, dx$$
or ...
- 5,337
4
votes
1
answer
187
views
Prove $\int_{0}^{\pi}\frac{u}{1-\cos u}\ln\frac{1+\sin u}{1-\sin u}{d}u=\left(\pi+2\ln2\right)\pi$ [closed]
How to prove
$$\displaystyle\int_{0}^{\pi}\frac{u}{1-\cos u}\ln\left(\frac{1+\sin u}{1-\sin u}\right)\mathrm{d}u=\left(\pi+2\ln2\right)\pi\,\,?$$
I tried to apply the Feynman method to get the ...
- 907
0
votes
1
answer
71
views
Computing a singular integral
I would like to derive the exact value of the following integral
$$ I_s= \int_1^\infty r| r^{-2s}- (r^2-1)^{-s}|d r\qquad \text{with}\qquad 0<s<1.$$
The existence of $I_s$ is warranted since ...
- 24.2k
0
votes
3
answers
282
views
Integral of sinc function over bounded interval
I am trying to determine whether the integral
$$\int_0^1 \frac{\sin x}{x} dx$$ can be calculated analytically.
I am aware of the definition of the sine integral function $\text{Si}(x)$, but I haven't ...
- 332