3 studs

Calculus Level 4

$\large \int_{0}^{\pi /2} e^x\sin x\cos x \, dx$

If the above integral can be expressed as $\large \dfrac{a+{e^{{b\pi} / {c}}}}{d} ,$ where $a,b,c$ and $d$ are positive integers with $b$ and $c$ coprime, find the value of $a+b+c+d$ .