$\int_{-1}^{1} \sin^3 x \cos^2 x \, dx = $

The value of $ \int_{-3}^{3} (ax^5 + bx^3 + cx + k) dx $,where $a, b, c, k$ are constants,depends only on . . . . . .

For $n \in N$,the value of $\int_{0}^{n\pi + V} \sqrt{\frac{1 + \cos 2x}{2}} dx$ is . . . (where $\frac{\pi}{2} < V < \pi$)

The value of $\int_{-\pi}^{\pi} \frac{\cos^2 x}{1+\alpha^x} \, dx$ for $\alpha > 0$ is

Let $f: [2, 5] \to [2, 5]$ be a bijective function such that $\frac{d}{dx}(f^{-1}(x)) > 0$ for all $x \in [2, 5]$. Then $\int_{2}^{5} (f(x) + f^{-1}(x)) dx$ is

Let $N$ be the set of natural numbers. For $n \in N$,define $I_n = \int_0^\pi \frac{x \sin^{2n}(x)}{\sin^{2n}(x) + \cos^{2n}(x)} dx$. Then,for $m, n \in N$,which of the following is true?