3.8 Evaluation of integrals

1.

(a) $${\frac{1}{5}}$$(b) $-2 + 2 {\sqrt{2}}$(c) $${\frac{3 \pi }{16}}$$(d) $4\cos{1} - 2\sin{1}$ (e) 0
(f) $${\frac{\pi}{3} - \frac{\sqrt{3}}{2}}$$ (g) $1$

2.

$${I_{n} = \int_{0}^{\frac{\pi}{2}}\sin^{n}{x}dx = \int_{0}^{\frac{\pi}{2}}\sin^{n-1}{x} \sin{x}dx = \frac{n-1}{n} I_{n-2}}$$

$${J_{n} = \int_{0}^{\frac{\pi}{2}}\cos^{n}{x}dx = \int_{0}^{\frac{\pi}{2}}\cos^{n-1}{x} \cos{x}dx = \frac{n-1}{n} J_{n-2}}$$