Problem

Let

$$I_n = \int_0^{\pi/2} x^n \cos(x^2)\,dx \qquad n \ge 1.$$

(a) Prove that, for $n \ge 5$,

$$I_n = \frac{1}{2}\left(\frac{\pi}{2}\right)^{n-1} - \frac{(n-1)(n-3)}{4}I_{n-4}.$$

(b) Hence, determine the exact value of $I_5$, giving your answer in its simplest form.