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

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

$\int_0^{\pi /2} {\frac{1}{{1 + \sqrt {\tan x} }}} \,dx = $

If $I_{1} = \int_{0}^{1} (1 - x^{50})^{100} dx$ and $I_{2} = \int_{0}^{1} (1 - x^{50})^{101} dx$ such that $I_{2} = \alpha I_{1}$,then $\alpha$ is equal to:

For $x \in \mathbb{R}$,let $f(x) = |\sin x|$ and $g(x) = \int_0^x f(t) \, dt$. Let $p(x) = g(x) - \frac{2}{\pi} x$. Then:

Let $I_{n} = \int_{1}^{e} x^{19}(\log |x|)^{n} dx$,where $n \in N$. If $(20) I_{10} = \alpha I_{9} + \beta I_{8}$,for natural numbers $\alpha$ and $\beta$,then $\alpha - \beta$ is equal to ..... .