Question Number 160902 by blackmamba last updated on 08/Dec/21
$$\:\:\int\:\frac{{dx}}{\:\sqrt{\mathrm{sin}\:^{\mathrm{3}} {x}}\:\sqrt{\mathrm{cos}\:^{\mathrm{5}} {x}}}\:=? \\ $$
Answered by chhaythean last updated on 09/Dec/21
$$=\int\frac{\mathrm{dx}}{\:\sqrt{\frac{\mathrm{sin}^{\mathrm{3}} \mathrm{x}}{\mathrm{cos}^{\mathrm{3}} \mathrm{x}}×\mathrm{cos}^{\mathrm{8}} \mathrm{x}}} \\ $$$$=\int\frac{\mathrm{sec}^{\mathrm{4}} \mathrm{xdx}}{\mathrm{tan}^{\frac{\mathrm{3}}{\mathrm{2}}} \mathrm{x}} \\ $$$$\mathrm{let}\:\mathrm{u}=\mathrm{tanx}\Rightarrow\mathrm{du}=\mathrm{sec}^{\mathrm{2}} \mathrm{xdx} \\ $$$$=\int\frac{\mathrm{1}+\mathrm{u}^{\mathrm{2}} }{\mathrm{u}^{\frac{\mathrm{3}}{\mathrm{2}}} }\mathrm{du}=−\frac{\mathrm{2}}{\:\sqrt{\mathrm{u}}}+\frac{\mathrm{2}}{\mathrm{3}}\mathrm{u}^{\frac{\mathrm{3}}{\mathrm{2}}} +\mathrm{c} \\ $$$$=−\frac{\mathrm{2}}{\:\sqrt{\mathrm{tanx}}}+\frac{\mathrm{2}}{\mathrm{3}}\mathrm{tan}^{\frac{\mathrm{3}}{\mathrm{2}}} \mathrm{x}+\mathrm{c} \\ $$
Commented by peter frank last updated on 09/Dec/21
$$\mathrm{great} \\ $$