Let $\mathrm{a}$ and $\mathrm{b}$ be real constants such that the function $f$ defined by $f(x)=\left\{\begin{array}{cc}x^2+3 x+a & x \leq 1 \\ b x+2, & x>1\end{array}\right.$ be differentiable on $R$. Then, the value of $\int_{-2}^2 f(x) d x$ equals

Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function such that $\int_0^1 f(x) d x=10$. Which of the following statements is NOT necessarily true?

If $\frac{d}{{dx}}\,G\left( x \right) = \frac{{{e^{\tan \,x}}}}{x},\,x \in \left( {0,\pi /2} \right)$, then $\int\limits_{1/4}^{1/2} {\frac{2}{x}} .{e^{\tan \,\left( {\pi \,{x^2}} \right)}}dx$ is equal to

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 :

If $\int_{}^{} {f(x)\,dx} = x{e^{ - \log |x|}} + f(x),$ then $f(x)$ is

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,