Question Number 203544 by laxmanb2111 last updated on 21/Jan/24
$$\int_{\mathrm{0}} ^{\mathrm{1}_{} } \frac{{dx}}{\left(\mathrm{1}−{x}^{\mathrm{6}} \right)^{\frac{\mathrm{1}}{\mathrm{6}}} }\:=\frac{\pi}{\mathrm{3}} \\ $$
Answered by DwaipayanShikari last updated on 23/Jan/24
$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\left(\mathrm{1}−{x}^{{n}} \right)^{\mathrm{1}/{n}} \:}{dx} \\ $$$$=\frac{\mathrm{1}}{{n}}\int_{\mathrm{0}} ^{\mathrm{1}} {u}^{\frac{\mathrm{1}}{{n}}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{−\mathrm{1}/{n}} {du} \\ $$$$=\frac{\mathrm{1}}{{n}}{B}\left(\frac{\mathrm{1}}{{n}},\mathrm{1}−\frac{\mathrm{1}}{{n}}\right) \\ $$$$=\frac{\mathrm{1}}{{n}}\Gamma\left(\frac{\mathrm{1}}{{n}}\right)\Gamma\left(\mathrm{1}−\frac{\mathrm{1}}{{n}}\right)=\frac{\pi}{{nsin}\left(\pi/{n}\right)} \\ $$$${n}=\mathrm{6} \\ $$$$=\frac{\pi}{\mathrm{3}} \\ $$