$\int_{0}^{\pi / 2} \frac{\sin x-\cos x}{1-\sin x \cdot \cos x} d x$ is equal to

$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ is equal to :

$\int_0^\pi x \sin x \cos^4 x \, dx = $

$\int_0^{2n\pi } {\left( {|\sin x| - \left| {\frac{1}{2}\sin x} \right|} \right)} \;dx$ equals

Evaluate the following definite integral: $\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} d x$

Let $f:(0,2) \rightarrow R$ be defined as $f(x) = \log_{2}\left(1+\tan\left(\frac{\pi x}{4}\right)\right)$. Then,$\lim_{n \rightarrow \infty} \frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\ldots+f(1)\right)$ is equal to