Problem

Let $I_n = \int_0^1 (1 - x)^n\sinh x\,\mathrm{d}x$, where $n$ is a non-negative integer.

(a) Show that, for $n \ge 2$, $I_n = -1 + n(n - 1)I_{n - 2}$.

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