Question Number 124906 by Mammadli last updated on 06/Dec/20
$$\int\boldsymbol{{sinx}}^{\mathrm{3}} \boldsymbol{{dx}}=? \\ $$
Commented by Dwaipayan Shikari last updated on 06/Dec/20
$$\int{sinx}^{\mathrm{3}} {dx}\:\:\:\:\:\:\:\:\:\: \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{i}}\int{e}^{{ix}^{\mathrm{3}} } −{e}^{−{ix}^{\mathrm{3}} } {dx}\: \\ $$$$=\frac{{ix}\sqrt{{ix}^{\mathrm{3}} }}{\mathrm{6}{x}^{\mathrm{2}} }\Gamma\left(\frac{\mathrm{1}}{\mathrm{3}},{ix}^{\mathrm{3}} \right)+\frac{{ix}\sqrt{−{ix}^{\mathrm{3}} }}{\mathrm{6}{x}^{\mathrm{2}} }\Gamma\left(\frac{\mathrm{1}}{\mathrm{3}},−{ix}^{\mathrm{3}} \right)\:\left({Incomplete}\:{gamma}\right) \\ $$