$\int_{0}^{\pi /2} {\sin 2x \log \tan x \, dx}$ is equal to

$\int_{-\pi}^{\pi} |\pi - |x|| \, dx$ is equal to :

$\int_{0}^{\infty} \log \left( x + \frac{1}{x} \right) \frac{dx}{1 + x^2}$ is equal to

The value of the integral $\int_{-\pi}^{\pi} \frac{\cos^2 x}{1+a^x} dx$,where $a > 0$,is

$\int_0^1 {\log \sin \left( {\frac{\pi }{2}x} \right)} \,dx = $

Let $f: [0, \frac{\pi}{2}] \rightarrow [0, 1]$ be the function defined by $f(x) = \sin^2 x$ and let $g: [0, \frac{\pi}{2}] \rightarrow [0, \infty)$ be the function defined by $g(x) = \sqrt{\frac{\pi x}{2} - x^2}$.
(There are two questions based on this paragraph. The questions given below are those two.)
$(1)$ The value of $2 \int_0^{\frac{\pi}{2}} f(x) g(x) dx - \int_0^{\frac{\pi}{2}} g(x) dx$ is
$(2)$ The value of $\frac{16}{\pi^3} \int_0^{\frac{\pi}{2}} f(x) g(x) dx$ is