Problem

In $I_n=\int e^x\sin^n x\,dx \qquad n\in\mathbb Z,\ n\ge 0$

(a) Show that

$$I_n=\frac{e^x\sin^{n-1}x}{n^2+1}( \sin x-n\cos x )+\frac{n(n-1)}{n^2+1}I_{n-2} \qquad n\ge 2$$

(b) Hence find the exact value of

$$\int_0^{\pi/2}e^x\sin^4x\,dx$$

giving your answer in the form $Ae^\pi+B$ where $A$ and $B$ are rational numbers to be determined.