$\int_{-1}^{1} \log \left( \frac{1+x}{1-x} \right) \, dx = $

Let $\int_{-2}^{2} (\sin |x| + [x \sin x]) dx = 2(3 - \cos 2) + \beta$, where $[\cdot]$ denotes the greatest integer function. Then $\beta \sin \left(\frac{\beta}{2}\right)$ equals:

The value of $\sum_{n=1}^{10} \int_{-2n-1}^{-2n} \sin^{27} x \, dx + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx$ is equal to

The value of the integral $\int_0^{\frac{\pi}{2}} \frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} \,dx$ is

If $I_n = \int_0^{\pi / 4} \tan^n x \, dx$,then $I_2+I_4, I_3+I_5, I_4+I_6, \ldots$ are in

Let $a_n = \int_{-\pi}^{\pi} |x-1| \cos(nx) \, dx$ for all natural numbers $n$. Then,the sequence $(a_n)_{n \geq 1}$ satisfies: