3.8 定積分の計算

1.

(a) $\sin{x}$は奇関数で，積分範囲が$[-1,1]$より， $\int_{-1}^{1}\sin{x}\;dx = 0$

(b) $\sin^3{x}$は奇関数で，積分範囲が$[-1,1]$より， $\int_{-1}^{1}\sin^3{x}\;dx = 0$

(c)

$$\int_{0}^{\pi}\sin^3{x}\; dx = \frac{2!!}{3!!} = \frac{2}{3}$$

(d)

$$\int_{0}^{\pi}\cos{2x}\; dx = [\frac{\sin{2x}}{2}]_{0}^{\pi} = 0$$

(e)

$$\int_{-1}^{1}\sqrt{x+1}\; dx = \int_{-1}^{1}(x+1)^{1/2}\; dx = \l... ...2}{3}(x+1)^{3/2}\right]_{-1}^{1} = \frac{2}{3}\cdot 2^{3/2} = \frac{2^{5/2}}{3}$$