Problem

It is given that $I_n = \int_0^1 x^n\sinh x\,\mathrm{d}x$, where $n$ is a non-negative integer.

(a) Show that, for $n \ge 2$,

$$I_n = \cosh 1 - n\sinh 1 + n(n - 1)I_{n - 2}.$$

(b) Find the exact value of $I_2$.