If $[x]$ is the greatest integer $\leq x$, then $\pi^{2} \int_{0}^{2}\left(\sin \frac{\pi \mathrm{x}}{2}\right)(\mathrm{x}-[\mathrm{x}])^{[\mathrm{x}]} \mathrm{d} \mathrm{x}$ is equal to :

Let $\ln x$ denote the logarithm of $x$ with respect to the base $e$. Let $S \subset R$ be the set of all points where the function $\ln \left(x^2-1\right)$ is well-defined. Then, the number of functions $f: S \rightarrow R$ that are differentiable, satisfy $f^{\prime}(x)=\ln \left(x^2-1\right)$ for all $x \in S$ and $f(2)=0$, is

$I=\int \limits_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x$. Then

Let $f(x)=2+|x|-|x-1|+|x+1|, x \in R$. Consider

$(S1)$: $f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)+f^{\prime}\left(\frac{3}{2}\right)=2$

$( S 2): \int_{-2}^{2} f ( x ) dx =12$Then,

Let $\mathrm{f}$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1) .$ If $\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} \,d t=\int \limits_{0}^{x} f(t) \,d t$ $0 \leq x \leq 1$ and $f(0)=0$, then $\lim \limits _{x \rightarrow 0} \frac{1}{x^{2}} \int \limits_{0}^{x} f(t)\, d t:$