- #1

- 22,183

- 3,321

Integrals are pretty interesting, and there are a lot of different methods to solve them. In this thread, I will give as a challenge 10 integrals. Here are the rules:

Anyway, here we go:

- For a solution to count, the answer must not only be correct, but a detailed solution must also be given.
- A correct answer would consist of the correct number (which may be infinite) or the statement that the given integral does not exist.
- Any use of outside sources is allowed, but do not look up the integral directly. For example, it is ok to go check calculus textbooks for methods, but it is not allowed to type in the integral in wolframalpha.
- If you previously encountered this integral and remember the solution, then you cannot participate with that specific integral.
- All mathematical methods are allowed.

Anyway, here we go:

- [itex]\int_0^1 \frac{\text{ln}(1+x)}{1+x^2}dx = \frac{\pi}{4}\text{ln}(\sqrt{2})[/itex] SOLVED BY Ssnow
- [itex]\int_0^\infty \frac{|\sin(x)|}{x}dx = +\infty[/itex] SOLVED BY PeroK

- [itex]\int_0^\infty \frac{\text{atan}(2016x) - \text{atan}(1916x)}{x}dx = \frac{\pi}{2}\text{ln} \left(\frac{2016}{1916}\right)[/itex] SOLVED BY Samy_A

- [itex]\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}}dx = \frac{\pi}{4}[/itex] SOLVED BY Ssnow

- [itex]\int_0^1 \sqrt{-\text{ln}(x)}dx = \frac{\sqrt{\pi}}{2}[/itex] SOLVED BY Ssnow

- [itex]\int_0^1 \frac{1-4x^5}{(x^5 - x + 1)^2}dx = 1[/itex] SOLVED BY Samy_A
- [itex]\int_0^{\pi/2} \text{acos}\left(\frac{\cos(x)}{1+2\cos(x)}\right)dx[/itex]
- [itex]\int_0^\infty \frac{\sin^{9}(x)}{x}dx = \frac{35}{256}\pi[/itex] SOLVED BY vela
- [itex]\int_0^\infty \frac{x^{1916}}{x^{2016} + 1}dx = \frac{1}{2016} \pi \frac{1}{\sin{\frac{1917 \pi}{2016}}}[/itex] SOLVED BY fresh_42
- [itex]\int_{\sqrt{2}}^\infty \frac{1}{x + x^{\sqrt{2}}}dx = -\ln \frac{\sqrt{2}}{{(1+{\sqrt 2}^{\sqrt{2}-1}})^{\sqrt{2}+1}}[/itex] SOLVED BY Samy_A

Last edited: