1/(1+cos x)의 부정적분

$\displaystyle \frac{1}{1 + \cos x} = \frac{1 - \cos x}{( 1 + \cos x ) ( 1 - \cos x )} = \frac{1 - \cos x}{\sin ^{2} x} = \csc ^{2} x - \csc x \cot x$이므로 $\displaystyle f ( x ) = \int ( \csc ^{2} x - \csc x \cot x ) dx = - \cot x + \csc x + C$ ($C$는 적분상수) $\displaystyle f \left( \frac{\pi}{2} \right) = 0 + 1 + C = 1$에서 $C = 0$ $f ( x ) = - \cot x + \csc x$ 따라서 $\displaystyle f \left( \frac{\pi}{3} \right) = - \frac{1}{\sqrt{3}} + \frac{2}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ ①