Question Number 141713 by rs4089 last updated on 22/May/21
Answered by Dwaipayan Shikari last updated on 22/May/21
$$\mathrm{1}−{x}={u} \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{{u}} {x}^{\mathrm{1}/\mathrm{3}} {y}^{−\mathrm{1}/\mathrm{2}} \left({u}−{y}\right)^{\mathrm{1}/\mathrm{2}} {dydx}\:\:\:\:{y}={uk} \\ $$$$=\int_{−\mathrm{1}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{1}/\mathrm{3}} {uk}^{−\mathrm{1}/\mathrm{2}} \left(\mathrm{1}−{k}\right)^{\mathrm{1}/\mathrm{2}} {dkdx} \\ $$$$=\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\int_{−\mathrm{1}} ^{\mathrm{1}} {x}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}−{x}\right){dx}=\frac{\pi}{\mathrm{2}}\left[\frac{\mathrm{3}}{\mathrm{4}}{x}^{\mathrm{4}/\mathrm{3}} −\frac{\mathrm{3}}{\mathrm{7}}{x}^{\mathrm{7}/\mathrm{3}} \right]_{−\mathrm{1}} ^{\mathrm{1}} \\ $$$$=−\frac{\mathrm{3}\pi}{\mathrm{7}} \\ $$