Question

If $f\left(x\right)=\{\begin{array}{cc}{e}^{cosx}sinx,& for|x|\le 2,\\ 2,& otherwise\end{array},$ then ${\int }_{-2}^{3}f\left(x\right)dx=$

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is C 2

$f\left(x\right)=\{\begin{array}{cc}{e}^{cosx}sinx,& for|x|\le 2,\\ 2,& otherwise\end{array},$ ...(1)
$I={\int }_{-2}^{3}f\left(x\right)dx={\int }_{-2}^{2}f\left(x\right)dx+{\int }_2^{3}f\left(x\right)dx$
From (1)
$I={\int }_{-2}^2\left(e^{cosx}sinx\right)dx+{\int }_2^{3}2dx$
As $\left(e^{cosx}sinx\right)$ is odd function ${\int }_{-2}^2\left(e^{cosx}sinx\right)dx=0$
Therefore
$1=0+{\left[2x\right]}_2^3=6-4=2$