Evaluation of definite integrals

Exercise

1.

Evaluate the following definite integrals．

(a)

$${\int_{0}^{\frac{\pi}{2}}\cos^{4}{x}\sin{x}dx}$$

(b)

$${\int_{0}^{\frac{\pi}{2}}\frac{\sin{x}}{\sqrt{1+\cos{x}}} dx}$$

(c)

$${\int_{0}^{\frac{\pi}{2}}\sin^{4}{x}dx}$$

(d)

$${\int_{-1}^{1}x^2 \cos{x} dx}$$

(e)

$${\int_{0}^{\pi}\cos{nx}dx}$$ (n integer)

2.

$${\int_{0}^{\frac{\pi}{2}}\sin^{n}{x}dx = \int_{0}^{\frac{\pi}{2}}\cos^{n}{x}dx}$$ を示そう．