第9章 9.3

(1)

$\int_0^{\frac{\pi}{2}} \cos^2 x\,dx = \frac{1}{2}\cdot\frac{\pi}{2} = \frac{\pi}{4}$

(2)

$\int_1^2 \sin^2 \pi x\,dx$

$2\pi x = t$ とおくと $2\pi dx = dt$ $x:1\to 2$ $t:2\pi\to 4\pi$ これから

$=\int_{2\pi}^{4\pi}\frac{1}{2\pi}\sin^2\frac{1}{2}t\,dt=\frac{1}{2\pi}\int_{2\pi}^{4\pi}\frac{1-\cos t}{2}\,dt=\frac{1}{2\pi}\cdot\frac{1}{2}\left[t-\sin t\right]_{2\pi}^{4\pi}=\frac{1}{4\pi}(4\pi-0-2\pi+0)=\frac{1}{2}$

(3)

$\int_0^2 xe^{\frac{x}{2}}dx=\left[2xe^{\frac{x}{2}}\right]_0^2-2\int_0^2 e^{\frac{x}{2}}dx=2\cdot2\cdot e-2\cdot2\left[e^{\frac{x}{2}}\right]_0^2=4e-4(e-1)=4$

(4)

$\int_1^3 \log x\,dx=\left[x\log x-x\right]_1^3=3\log 3-3+1=3\log 3-2$