Problem

Let

$$I_n = \int x^n \cos x\,dx.$$

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

$$I_n = x^n\sin x + nx^{n-1}\cos x - n(n-1)I_{n-2}.$$

(b) Hence find the functions $f(x)$ and $g(x)$ such that

$$\int x^4\cos x\,dx = f(x)\sin x + g(x)\cos x + c,$$

where $c$ is an arbitrary constant.