$\int_0^\pi (\sin^5 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^4 x) dx =$

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^2 x \cos^2 x(\sin x + \cos x) dx =$

If $f(5-x)=f(x)$ and $\int_2^3 f(x) dx=2$,then $\int_2^3 x f(x) dx=$

$\int_{ - \pi }^{\pi } {\frac{{2x(1 + \sin x)}}{{1 + {{\cos }^2}x}}dx} $ is

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_{\frac{-\pi}{2}}^{\frac{\pi}{2}}(x^5-x^3 \cos x+\sin^3 x-3) \, dx = $ . . . . . .