Evaluate the definite integral: $\int_0^\pi \frac{x \cos^2 x}{1+\sin x} dx$

$\int_{-1}^1 (a x^3 + b x) dx = 0$ for

Let $I = \int_{0}^{1} \frac{x^{3} \cos 3x}{2+x^{2}} dx$. Then

$\int_{0}^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} = $

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

Let $T > 0$ be a fixed number. $f: R \rightarrow R$ is a continuous function such that $f(x+T) = f(x)$ for all $x \in R$. If $I = \int_0^T f(x) dx$,then find the value of $\int_0^{5T} f(2x) dx$.