Question

If $f\left(x\right)=\{\begin{array}{cc}{e}^{\cos x}\cdot \sin x& for\left|x\right|\le 2\\ 2& otherwise\end{array}$ then ${\int }_{-2}^{3}f\left(x\right)dx$ is equal to

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is C 2

Given $f\left(x\right)=\{\begin{array}{c}{e}^{\cos x}\sin x,\left|x\right|\le 2\\ 2else\end{array}$
$\therefore {\int }_{-2}^{2}f\left(x\right)dx+{\int }_2^{3}f\left(x\right)dx={\int }_{-2}^2{e}^{\cos x}\sin xdx+{\int }_2^{3}2dx$
$=0+2{\left[x\right]}_2^3$ ($\because e^{\cos x}\sin x$ is an odd function)
$=2[3-2]=2[\because {\int }_{-2}^{3}f\left(x\right)dx=2]$