Let $f$ be a positive function. Let $I_1 = \int_{1 - k}^k x f\{x(1 - x)\} dx$ and $I_2 = \int_{1 - k}^k f\{x(1 - x)\} dx$,where $2k - 1 > 0$. Then $I_1/I_2$ is

$\int_{-\pi/2}^{\pi/2} \frac{\sin^2 x}{1 + 2^x} dx = \dots$

If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^2 \cos^2 x}{1+e^x} dx = \pi(\alpha \pi^2 + \beta)$,where $\alpha, \beta \in \mathbb{Z}$,then $(\alpha + \beta)^2$ equals:

The value of $\int_{-1}^{3} \left( \tan^{-1} \left( \frac{x}{x^2+1} \right) + \tan^{-1} \left( \frac{x^2+1}{x} \right) \right) dx$ is

If $\int_{-1}^{4} f(x) dx = 4$ and $\int_{2}^{4} (3 - f(x)) dx = 7$,then the value of $\int_{2}^{-1} f(x) dx$ 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