Problem

The integral $I_n$ is defined by

$$I_n = \int_0^1 \big(1 + x^2\big)^n \,\mathrm{d}x.$$

(a) By considering $\frac{\mathrm{d}}{\mathrm{d}x}\big(x(1 + x^2)^n\big)$, or otherwise, show that

$$(2n + 1)I_n = 2^n + 2nI_{n - 1}.$$

(b) Find the exact value of $I_{-2}$.