第10章 10.3

(1)

$\int_0^{2\pi} 2\sin\frac{x}{2}\,dx = 2\left[-2\cos\frac{x}{2}\right]_0^{2\pi} = -4(-1-1) = 8$

(2)

$\int_{\frac{\pi}{4}}^{\frac{5}{4}\pi} (\sin x - \cos x)\,dx = \left[-\cos x - \sin x\right]_{\frac{\pi}{4}}^{\frac{5}{4}\pi} = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}$

(3)

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \tan x\,dx = \left[-\log|\cos x|\right]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = -\log\frac{1}{2} + \log\frac{\sqrt{3}}{2} = \log\sqrt{3}$

(4)

$\int_0^{\frac{\pi}{4}} (\tan x - \sin x)\,dx = \left[-\log|\cos x| + \cos x\right]_0^{\frac{\pi}{4}} = -\log\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} - 1 = \log\sqrt{2} + \frac{\sqrt{2}-2}{2}$