3.8 Evaluation of integrals

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

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

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