If $\int_0^{\pi / 2} \tan ^{14}\left(\frac{x}{2}\right) d x=2\left[\sum_{n=1}^7 f(n)-\frac{\pi}{4}\right]$, then $f(n)=$

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

For $n \in N$,if $I_n = \int \frac{\sin nx}{\sin x} dx = \frac{2}{n-1} \sin(n-1)x + I_{n-2}$ and $\int_0^\pi \frac{\sin nx}{\sin x} dx = \frac{k\pi}{2}$,then $k =$

For any real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10, 10]$ by $f(x) = \begin{cases} x - [x], & \text{if } [x] \text{ is odd} \\ 1 + [x] - x, & \text{if } [x] \text{ is even} \end{cases}$. Then the value of $\frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos(\pi x) dx$ is:

For $0 \le x \le \frac{\pi}{2}$,the value of $\int_{0}^{\sin^{2}x} \sin^{-1}(\sqrt{t}) \, dt + \int_{0}^{\cos^{2}x} \cos^{-1}(\sqrt{t}) \, dt$ is equal to

$\int_0^\pi \sin^2 x \cos^3 x \, dx = $ . . . . . . .