第5章 5.4

(1)

$\int_0^{\pi/2}\sin^5 x\,dx=\frac{4}{5}\cdot\frac{2}{3}=\frac{8}{15}$

(2)

$\int_0^{\pi/2}\cos^6 x\,dx=\frac{5}{6}\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}=\frac{5}{32}\pi$

(3)

$\int_0^{\pi/2}\sin^2 x\cdot\cos^6 x\,dx=\int_0^{\pi/2}\cos^6 x-\cos^8 x\,dx=\frac{5}{6}\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}-\frac{7}{8}\cdot\frac{5}{6}\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}=\frac{5}{256}\pi$

(4)

$\int_0^{\pi/2}\sin^4 x\cdot\cos^3 x\,dx=\int_0^{\pi/2}(1-\cos^2 x)^2\cdot\cos^3 x\,dx=\int_0^{\pi/2}(1-2\cos^2 x+\cos^4 x)\cdot\cos^3 x\,dx$

$=\int_0^{\pi/2}(\cos^3 x-2\cos^5 x+\cos^7 x)dx=\frac{2}{3}-2\cdot\frac{4}{5}\cdot\frac{2}{3}+\frac{6}{7}\cdot\frac{4}{5}\cdot\frac{2}{3}=\frac{2}{35}$

(5)

$\int_0^{\pi}\cos^3 x\,dx=\int_0^{\pi}\cos x\cdot\cos^2 x\,dx=\int_0^{\pi}\cos x\cdot(1-\sin^2 x)dx=\int_0^{\pi}(\cos x-\cos x\cdot\sin^2 x)dx$

$=\int_0^{\pi}\cos x\,dx-\int_0^{\pi}\cos x\cdot\sin^2 x\,dx \qquad \sin x=t\ \text{とおくと} \qquad \cos x\,dx=dt \qquad x:0\to\pi \qquad t:0\to0$

これから

$=\int_0^{\pi}\cos x\,dx-0=[\sin x]_0^{\pi}=0$

(6)

$\int_0^{2\pi}\sin^8 x\,dx=2\int_0^{\pi}\sin^8 x\,dx=4\int_0^{\pi/2}\sin^8 x\,dx=4\cdot\frac{7}{8}\cdot\frac{5}{6}\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}=\frac{35}{64}\pi$