Question

lf $f\left(x\right)=\{\begin{array}{c}{e}^{\cos x}\sin x,for|x|\le 2\\ 2;otherwise\end{array}$, then ${\int }_{-2}^{3}f\left(x\right)dx$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

$I={\int }_{-2}^{3}f\left(x\right)dx$

$I={\int }_{-2}^2{e}^{\cos x}\sin xdx+{\int }_2^{3}2dx$

consider, $f\left(x\right)=e^{\cos x}\sin x$

so, $f(-x)=e^{\cos (-x)}\sin (-x)$

$\Rightarrow$ $f(-x)=-e^{\cos x}\sin x=-f\left(x\right)$

$\Rightarrow$ $e^{\cos x}\sin x$ is an odd function

hence, ${\int }_{-2}^2{e}^{\cos x}\sin xdx=0$

Thus, $I={\int }_2^{3}2dx$

$I=2{\left[x\right]}_2^3=2(3-2)$

$I=2$

Hence, answer is option-(C).