The value of the integral $\frac{48}{\pi^{4}} \int_{0}^{\pi} \left(\frac{3 \pi x^{2}}{2} - x^{3}\right) \frac{\sin x}{1 + \cos^{2} x} dx$ is equal to

$\int_0^2 x^8\left(\frac{4}{x^2}-1\right)^{5 / 2} d x=$

$\int_0^{\pi / 2} \frac{200 \sin x+100 \cos x}{\sin x+\cos x} d x$ is equal to (in $\pi$)

If ${I_n} = \int_0^{\pi /4} {{\tan ^n}\theta \,d\theta }$,then ${I_8} + {I_6}$ equals

Read the following mathematical statements carefully:
$I.$ $A$ differentiable function $f$ with maximum at $x = c$ $\implies f''(c) < 0$.
$II.$ Antiderivative of a periodic function is also a periodic function.
$III.$ If $f$ has a period $T$ then for any $a \in R$,$\int\limits_0^T {f(x)\,dx} = \int\limits_0^T {f(x + a)\,dx}$.
$IV.$ If $f(x)$ has a maxima at $x = c$,then $f$ is increasing in $(c - h, c)$ and decreasing in $(c, c + h)$ as $h \to 0$ for $h > 0$. Now indicate the correct alternative.

If $n$ is a positive integer and $[x]$ is the greatest integer not exceeding $x$,then $\int_0^n {\{x - [x]\} \,dx}$ equals