Question

The value of $\underset{0}{\overset{\mathrm{\pi }/2}{\int }}\cos x{e}^{\sin x}dx$ is
(a) 1
(b) e − 1
(c) 0
(d) − 1

Answer 1

(b) e − 1


$$\begin{gathered} \mathrm{Let},I={\int }_0^{\frac{\mathrm{\pi }}{2}}\cos x{e}^{\sin x}dx \\ \mathrm{Let}\sin x=t,\mathrm{then}\cos xdx=dt \\ \mathrm{When}x=0,t=0\mathrm{and}x=\frac{\mathrm{\pi }}{2},t=1 \\ \mathrm{Therefore}\mathrm{the}\mathrm{integral}\mathrm{becomes} \\ I={\int }_0^1{\mathrm{e}}^{\mathrm{t}}\mathrm{dt} \\ ={\left[{\mathrm{e}}^{\mathrm{t}}\right]}_0^1 \\ =\mathrm{e}-1 \end{gathered}$$