If $\mathrm{f}(x)=\left\{\begin{array}{ll}\mathrm{e}^{\cos x} \sin x & , \text { for }|x| \leq 2 \\ 2, &…
If $\mathrm{f}(x)=\left\{\begin{array}{ll}\mathrm{e}^{\cos x} \sin x & , \text { for }|x| \leq 2 \\ 2, & \text { otherwise }\end{array}\right.$, then $\int_{-2}^3 \mathrm{f}(x) \mathrm{d} x$ is equal to
0
2
1
3
Solution
$\begin{aligned}
\int_{-2}^3 \mathrm{f}(x) \mathrm{d} x & =\int_{-2}^2 \mathrm{f}(x) \mathrm{d} x+\int_2^3 \mathrm{f}(x) \mathrm{d} x \\
& =\int_{-2}^2 \mathrm{e}^{\cos x} \sin x \mathrm{~d} x+\int_2^3 2 \mathrm{~d} x
\end{aligned}$
Since $\mathrm{e}^{\cos x} \sin x$ is an odd function.
$\therefore \quad \int_{-2}^3 \mathrm{f}(x) \mathrm{d} x=0+2(3-2)=2$