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sinx-3-dx-




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

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