$\int_{\frac{\pi}{5}}^{\frac{3 \pi}{10}} \frac{\tan x}{\tan x + \cot x} \, dx =$

If $\int_{ - a}^a {\sqrt {\frac{{a - x}}{{a + x}}} \,dx = k\pi ,} $ then $k = $

Let $[x]$ denote the largest integer not exceeding $x$ and $\{x\}=x-[x]$. Then,$\int \limits_0^{2012} \frac{e^{\cos (\pi\{x\})}}{e^{\cos (\pi\{x\})}+e^{-\cos (\pi\{x\})}} d x$ is equal to

$\int_0^\pi \frac{\theta \sin \theta}{1+\cos ^2 \theta} d \theta$ is equal to

Let $\int\limits_0^1 {{{\tan }^{ - 1}}\left( {\frac{{\tan x}}{2}} \right)} dx = \alpha $. Then $\int\limits_0^1 {{{\tan }^{ - 1}}\left( {\frac{{\tan x - 2\cot x}}{3}} \right)} dx$ is equal to:

If $M=\int_0^{\infty} \frac{\log t}{1+t^3} d t$ and $N=\int_{-\infty}^{\infty} \frac{t e^{2 t}}{1+e^{3 t}} d t$,then