Answer
$\frac{e^{sin \left(\frac{\pi^{2}}{2}\right)}-1}{\pi}$
Work Step by Step
Solve for the indefinite integral
$\int e^{sin \pi x} cos(\pi x)dx$
let $ sin (\pi x) =u$
$\pi cos(\pi x) dx=du$
$ cos(\pi x) dx=\frac{du}{\pi}$
$\int e^{sin \pi x} cos(\pi x)dx$
$=\frac{1}{\pi} \int e^udu$
$=\frac{1}{\pi}e^u+C$
$=\frac{1}{\pi}e^{sin (\pi x)}+C$
Solve for the definite integral
$\int e^{sin \pi x} cos(\pi x)dx$ $[0, \frac{\pi}{2}]$
$=\frac{1}{\pi}e^{sin (\pi x)}$ $[0, \frac{\pi}{2}]$
$=\frac{1}{\pi}e^{sin (\frac{\pi^{2}}{2} )}-\frac{1}{\pi}e^{sin (0)}$
$=\frac{e^{sin (\frac{\pi^{2}}{2})}-1}{\pi}$
