$\int_0^\pi x f(\sin x) dx = $

$\int_{3}^{5} \frac{\sqrt{x}}{\sqrt{8-x}+\sqrt{x}} \, dx =$

If $f$ is a continuous function and $f(x+T)=f(x)$ for all $x \in R$,it is given that $\int_0^{NT} f(t) dt = N \int_0^T f(t) dt$ (where $N$ is a natural number). Then,evaluate $\int_0^{50\pi} \sqrt{1-\cos 2x} dx$.

If $I_n = \int_0^{\pi / 2} \sin^n(x) dx$ and $I_n = (k) I_{n-2}$,then what will be the value of $k$?

$\int_{0}^{\pi} \cos^3 x \, dx = $

Suppose $M = \int_{0}^{\pi / 2} \frac{\cos x}{x+2} dx$ and $N = \int_{0}^{\pi / 4} \frac{\sin x \cos x}{(x+1)^{2}} dx$. Then, the value of $(M - N)$ equals