치환적분

$\displaystyle 0 \leq x \leq \frac{\pi}{2}$에서 $\sin x \geq 0$, $\cos x \geq 0$이므로 $\displaystyle \int _{0} ^{\frac{\pi}{2}} \sqrt{\sin x - \sin ^{3} x} dx$$\displaystyle = \int _{0} ^{\frac{\pi}{2}} \sqrt{\sin x \left( 1 - \sin ^{2} x \right)} dx$ $\displaystyle = \int _{0} ^{\frac{\pi}{2}} \sqrt{\sin x \cos ^{2} x} dx$ $\displaystyle = \int _{0} ^{\frac{\pi}{2}} \left| \cos x \right| \sqrt{\sin x} dx$ $\displaystyle = \int _{0} ^{\frac{\pi}{2}} {} \cos x \sqrt{\sin x} dx$ $\cdots$$\cdots$ ㉠ ㉠에서 $\sin x = t$로 놓으면 $x = 0$일 때, $t = 0$, $\displaystyle x = \frac{\pi}{2}$일 때, $t = 1$이고 $\displaystyle \cos x = \frac{dt}{dx}$이므로 $\displaystyle \int _{0} ^{\frac{\pi}{2}} \cos x \sqrt{\sin x} dx$$\displaystyle = \int _{0} ^{1} \sqrt{t} dt$$\displaystyle = \int _{0} ^{1} t ^{\frac{1}{2}} dt$$\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} \frac{2}{3} t ^{\frac{3}{2}} \end{array} \right] _{0} ^{1}$ $\displaystyle = \frac{2}{3} - 0 = \frac{2}{3}$