cos-y-3-dy-

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

\int c o s (y^{3}) d y

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

\int c o s (y^{3}) d y

= \frac{1}{2} \int e^{{i y}^{3}} + e^{- {i y}^{3}} d y y^{3} = i u

= \frac{i}{2} \int y^{- 2} e^{- u} d u - \frac{i}{2} \int y^{- 2} e^{- j} d j y^{3} = - i j

= \frac{i}{6} \int {(i u)}^{- \frac{2}{3}} e^{- u} d u - \frac{i}{2} \int {(- i u)}^{- \frac{2}{3}} e^{- j} d j

= \frac{\sqrt[3]{i}}{6} Γ (\frac{1}{3}, u) + \frac{\sqrt[3]{- i}}{6} Γ (\frac{1}{3}, - u) (I n c o m p l e t e G a m m a)

= \frac{\sqrt[3]{i}}{6} Γ (\frac{1}{3}, {i x}^{3}) + \frac{\sqrt[3]{- i}}{6} Γ (\frac{1}{3}, - {i x}^{3})