$\int_0^1 f(1 - x) \, dx$ has the same value as which of the following integrals?

If $I_1 = \int_0^{3 \pi} f(\cos^2 x) dx$ and $I_2 = \int_0^\pi f(\cos^2 x) dx$, then

If the value of the integral $\int_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$ is $\frac{2}{\pi}$,then a value of $\alpha$ is

If $\alpha = 1$ and $\beta = 1 + i\sqrt{2}$, where $i = \sqrt{-1}$ are two roots of the equation $x^3 + ax^2 + bx + c = 0$ with $a, b, c \in R$, then $\int_{-1}^{1} (x^3 + ax^2 + bx + c) dx$ is equal to:

$\int_{\frac{\pi}{18}}^{\frac{4\pi}{9}} \frac{2\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx = . . . . . .$

If $I_{m, n} = \int_{0}^{1} x^{m-1}(1-x)^{n-1} dx$ for $m, n \geq 1$ and $\int_{0}^{1} \frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}} dx = \alpha I_{m, n}$,where $\alpha \in R$,then $\alpha$ equals .... .