第9章 9.4

(1)

$\int_{-2}^{2} (x + 3x^3 - 5x^5)\,dx = \left[\frac{1}{2}x^2 + \frac{3}{4}x^4 - \frac{5}{6}x^6\right]_{-2}^{2} = 2 + 12 - \frac{80}{3} - 2 - 12 + \frac{80}{3} = 0$

(2)

$\int_{-3}^{3} (x^7 - 5x^5 + 3x^3 + 3x^2 - 7x - 4)\,dx = \left[\frac{1}{8}x^8 - \frac{5}{6}x^6 + \frac{3}{4}x^4 + x^3 - \frac{7}{2}x^2 - 4x\right]_{-3}^{3} = 54 - 24 = 30$

(3)

$\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \sin^3\theta\, d\theta = \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \sin\theta \cdot (1-\cos^2\theta)\,d\theta = \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \sin\theta - \sin\theta\cdot\cos^2\theta\, d\theta$

$= \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \sin\theta\, d\theta - \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \sin\theta\cdot\cos^2\theta\, d\theta \qquad \text{第2\text{項を} } \cos\theta = t \text{ として}$

$= [-\cos\theta]_{\frac{\pi}{2}}^{\frac{3\pi}{2}} + \int_0^0 t^2\,dt = 0$

(4)

$\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos^3\theta\, d\theta \qquad \cos\theta = \sin\left(\theta+\frac{\pi}{2}\right) \text{ なので与式は}$

$\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \sin^3\left(\theta+\frac{\pi}{2}\right) d\theta \qquad \theta+\frac{\pi}{2}=x \text{ とおくと} \qquad \theta: \frac{\pi}{2}\to\frac{3\pi}{2} \qquad x:\pi\to 2\pi \qquad \text{これから}$

$\int_{\pi}^{2\pi} \sin^3 x\, dx = -2\int_0^{\frac{\pi}{2}} \sin^3 x\, dx = -2\cdot\frac{2}{3} = -\frac{4}{3}$

(5)

$\int_{\frac{\pi}{4}}^{-\frac{\pi}{4}} \tan^3\theta \, d\theta$

tanθ は原点を中心とした奇関数であり、奇関数 × 奇関数 × 奇関数 ＝ 奇関数 から

$\int_{\frac{\pi}{4}}^{-\frac{\pi}{4}} \tan^3\theta \, d\theta = 0$