All Questions
6
questions
12
votes
2
answers
718
views
General expressions for $\mathcal{L}(n)=\int_{0}^{\infty}\operatorname{Ci}(x)^n\text{d}x$
Define $$\operatorname{Ci}(x)=-\int_{x}^{
\infty} \frac{\cos(y)}{y}\text{d}y.$$
It is easy to show
$$
\mathcal{L}(1)=\int_{0}^{\infty}\operatorname{Ci}(x)\text{d}x=0
$$
and
$$\mathcal{L}(2)=\int_{0}^{\...
- 5,370
4
votes
3
answers
211
views
Does $\int_0^{2\pi}\frac{d\phi}{2\pi} \,\ln\left(\frac{\cos^2\phi}{C^2}\right)\,\ln\left(1-\frac{\cos^2\phi}{C^2}\right)$ have a closed form?
I am wondering if anyone has a nice way of approaching the following definite integral $\newcommand{\dilog}{\operatorname{Li}_2}$
$$\int_0^{2\pi}\frac{d\phi}{2\pi} \,\ln\left(\frac{\cos^2\phi}{C^2}\...
- 179
6
votes
2
answers
198
views
Computing closed-form of $\int_0^{\infty}\frac{\arctan x}{a^2x^2+1}\,dx$
Find the close form of the integral
$$\int_0^{\infty}\frac{\arctan x}{a^{2}x^2+1}\,dx,\qquad a > 0.$$
I think this integral related with polylogarithm function.
My attempt as follows:
Let $$I(b)=\...
- 751
5
votes
5
answers
313
views
How can I compute this integral in closed form : $\int_0^{\frac{π}{4}}\ln^2(\tan x)dx$
How can I compute this integral in closed form :
$$\displaystyle\int_{0}^{\displaystyle \tfrac{π}{4}}\ln^{2}\left(\tan x\right)dx$$
How can use Fourier series here ?
$$-2\displaystyle \sum_{n=0}^{\...
user670794
0
votes
0
answers
46
views
Evaluate a certain one-dimensional integral involving inverse trigonometric functions
Demonstrate that the integral of
\begin{equation}
\cos (y) \left(\sqrt{4-\sin ^2(y)} \cos ^{-1}(\sin (y))+4 \cos (y) \csc ^{-1}(2 \csc
(y))\right)
\end{equation}
over $y \in [0,\frac{\pi}{2}]$ ...
- 513
8
votes
3
answers
745
views
Prove $\int_{\frac{\pi}{20}}^{\frac{3\pi}{20}} \ln \tan x\,\,dx= - \frac{2G}{5}$
Context:
This question
asks to calculate a definite integral which turns out to be equal to $$\displaystyle 4 \, \text{Ti}_2\left( \tan \frac{3\pi}{20} \right) -
4 \, \text{Ti}_2\left( \tan \frac{\pi}{...
- 6,645