$\int \limits_{6}^{16} \frac{\log _{e} x^{2}}{\log _{e} x^{2}+\log _{e}\left(x^{2}-44 x+484\right)} d x$ is equal to:

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

The value of $\int_0^{\pi / 2} \frac{d x}{1+(\tan x)^{\sqrt{2018}}}$ is equal to

The value of the integral $\int_{4}^{10} \frac{[x^2]}{[(x-14)^2] + [x^2]} dx$,where $[x]$ denotes the greatest integer function,is

$ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{dx}{e^{\sin x}+1} $ is equal to

If $f : R \rightarrow R$ is a continuous function satisfying $\int \limits_0^{\pi / 2} f(\sin 2x) \cdot \sin x \, dx + \alpha \int \limits_0^{\pi / 4} f(\cos 2x) \cdot \cos x \, dx = 0$,then $\alpha$ is equal to