question_answer

If \[f(x)=\left\{ \begin{align}   & {{e}^{\cos x}}\,\sin x,\,for\,|x|\le 2 \\  & 2,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise \\ \end{align} \right\}\]. Then  \[\int\limits_{-2}^{3}{f\,(x)\,\,dx=}\]

A) 0         

B) 1          

C) 2                     

D) 3

Correct Answer: C

Solution :

If

=\[\int\limits_{-2}^{3}{f(x)=dx=\int\limits_{-2}^{2}{f(x)\,\,dx+\int\limits_{2}^{3}{f(x)\,\,dx}}}\]

\[=\int\limits_{-2}^{2}{{{e}^{\cos x}}\sin \,x\,dx+\int\limits_{2}^{3}{2\,dx=0+2\,\left[ x \right]_{2}^{3}}}\]

\[\left[ \because \,\,{{e}^{\cos x}}\sin x\,\text{is an odd function} \right]\]

\[=2\left[ 3-2 \right]=2\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\int\limits_{-2}^{3}{f\left( x \right)\,\,dx=2}\]