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Question Number 129519 by BHOOPENDRA last updated on 16/Jan/21

$$\int cos\left(y^3\right)dy$$

Answered by Dwaipayan Shikari last updated on 16/Jan/21

$$\int cos\left(y^3\right)dy$$$$=\frac{1}{2}\int e^{{iy}^3}+e^{-{iy}^3}dy{y}^3=iu$$$$=\frac{i}{2}\int y^{-2}{e}^{-u}du-\frac{i}{2}\int y^{-2}{e}^{-j}dj{y}^3=-ij$$$$=\frac{i}{6}\int {\left(iu\right)}^{-\frac{2}{3}}{e}^{-u}du-\frac{i}{2}\int {(-iu)}^{-\frac{2}{3}}{e}^{-j}dj$$$$=\frac{\sqrt[3]{i}}{6}\mathrm{\Gamma }(\frac{1}{3},u)+\frac{\sqrt[3]{-i}}{6}\mathrm{\Gamma }(\frac{1}{3},-u)\left(IncompleteGamma\right)$$$$=\frac{\sqrt[3]{i}}{6}\mathrm{\Gamma }(\frac{1}{3},{ix}^3)+\frac{\sqrt[3]{-i}}{6}\mathrm{\Gamma }(\frac{1}{3},-{ix}^3)$$

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