Question Number 198052 by Mastermind last updated on 09/Oct/23
Answered by Sutrisno last updated on 09/Oct/23
$${x}={asin}\theta\:\Rightarrow\:{dx}={acos}\theta{d}\theta \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {a}^{\mathrm{2}} {sin}^{\mathrm{2}} \theta\left({a}^{\mathrm{2}} −{a}^{\mathrm{2}} {sin}^{\mathrm{2}} \theta\right)^{−\frac{\mathrm{3}}{\mathrm{2}}} {acos}\theta{d}\theta \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {a}^{\mathrm{2}} {sin}^{\mathrm{2}} {x}\left({a}^{\mathrm{2}} {cos}^{\mathrm{2}} \theta\right)^{−\frac{\mathrm{3}}{\mathrm{2}}} {acos}\theta{d}\theta \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {a}^{\mathrm{2}} {sin}^{\mathrm{2}} {xa}^{−\mathrm{3}} {cos}^{−\mathrm{3}} \theta{acos}\theta{d}\theta \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {tan}^{\mathrm{2}} \theta{d}\theta \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {sec}^{\mathrm{2}} \theta−\mathrm{1}{d}\theta \\ $$$${tan}\theta−\theta\mid_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \\ $$$$\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}−\frac{\pi}{\mathrm{6}} \\ $$$$ \\ $$