All Questions
11
questions
1
vote
2
answers
148
views
Is there a closed form for the integral $\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$, where $k$ is a natural number?
To start, I am aware that our integral $I(k)=\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$ is equal to $$I(k)=\int_{0}^{\pi/4}x^{k}\ln\sin x\, dx-\int_{0}^{\pi/4}x^{k}\ln\cos x\, dx$$, but I cannot seem to ...
- 586
8
votes
3
answers
264
views
How to evaluate $\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, dx$?
Question: How to evaluate $$\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, dx?$$
My attempt
We will denote the main integral as $\Omega$.
$$\Omega=\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, ...
12
votes
1
answer
653
views
Prove $\int_0^\pi\arcsin(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}})dx=\frac{\pi^2}{5}$.
There is numerical evidence that
$$\int_0^\pi\arcsin\left(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}}\right)dx=\frac{\pi^2}{5}.$$
How can this be proved?
Context
In another question, three random ...
- 25.7k
20
votes
1
answer
1k
views
Prove $\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12$.
There is numerical evidence that
$$I=\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12.$$
How can this be proved?
...
- 25.7k
3
votes
1
answer
180
views
How to integrate $\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$
Question
How to integrate $$\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$$
My attempt
\begin{align*}I &= \int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^...
3
votes
1
answer
98
views
Need help finding a general formula for integrals involving powers of $\arctan$
I have been recently looking at integrals of the following form:
$$\int_0^{1} \frac{\arctan^n{ \left( x \right)}}{1+x}dx, n \in \mathbb{N}$$
Some interesting patterns seem to be emerging for the cases ...
27
votes
4
answers
1k
views
Compute close-form of $\int_0^{\frac\pi2}\frac{dt}{\sin t+\cos t+\tan t+\cot t+\csc t+\sec t}$
I came across the improper trigonometric integral recently shared in a Chinese web forum
\begin{align}
\int_0^{\frac\pi2}\frac{dt}{\sin t+\cos t+\tan t+\cot t+\csc t+\sec t}\\
\end{align}
which amuses ...
- 99.7k
0
votes
2
answers
256
views
Integral $\int_0^{\pi / 4}\frac{x\tan^2 x\ln\left(\tan x\right)}{\cos^2 x}dx$
Compute in closed form without using series:
$$I =\int_0^{\pi / 4}\frac{x\tan^2 x\ln\left(\tan x\right)}{\cos^2 x}dx$$
I thought of using: $y=\tan x$ then $dy=\frac{1}{\cos^2 x}$, so :
$$I =\...
user668815
6
votes
3
answers
2k
views
Closed form of $\int_0^{\pi/2}x^{n}\ln{(\sin x)}dx $
Does a closed form of the above integral exists? $$\displaystyle \int_0^{\pi/2}x^{n}\ln{(\sin x)}dx $$
where $n$ is a positive integer.
- 447
11
votes
3
answers
665
views
Closed form of $ \int _{ 0 }^{ \pi /2 }{ x\sqrt { \tan { x } } \log { (\cos { x } ) }\ dx }$
Does there exists a closed form of$$ \displaystyle \int _{ 0 }^{ \pi /2 }{ x\sqrt { \tan { x } } \log { (\cos { x } ) }\ dx }$$
If exists can someone find a way to tackle this integral and provide a ...
- 314
34
votes
5
answers
2k
views
Integrate $\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$
Please help me to solve this integral:
$$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$
I managed to calculate an indefinite integral of the left part:
$$\int\frac{\cos x}{\sin x}...
- 2,575