Question Number 212007 by Nadirhashim last updated on 26/Sep/24
$$\:\:\:\:\int\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)\:\sqrt[{\mathrm{3}}]{\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)}\:.\boldsymbol{{dx}} \\ $$
Answered by Ghisom last updated on 26/Sep/24
$$=\int\left(\mathrm{cos}\:{x}\right)^{−\mathrm{1}/\mathrm{3}} \left(\mathrm{sin}\:{x}\right)^{\mathrm{4}/\mathrm{3}} {dx}= \\ $$$$=\frac{\mathrm{3}}{\mathrm{7}}\:_{\mathrm{2}} {F}_{\mathrm{1}} \:\left(\frac{\mathrm{2}}{\mathrm{3}},\:\frac{\mathrm{7}}{\mathrm{6}},\:\frac{\mathrm{13}}{\mathrm{6}},\:\mathrm{sin}^{\mathrm{2}} \:{x}\right)\:\mathrm{sin}^{\mathrm{7}/\mathrm{3}} \:{x}\:+{C} \\ $$