What is the integral of y^3cos(2y) ?

The integral of y^3cos(2y) is 1/2 y^3sin(2y)-3/2 (-1/2 y^2cos(2y)+1/4 (2ysin(2y)+cos(2y)))+C

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integral of y^3cos(2y)

\int y^{3} cos (2 y) d y

\frac{1}{2} y^{3} sin (2 y) - \frac{3}{2} (- \frac{1}{2} y^{2} cos (2 y) + \frac{1}{4} (2 y sin (2 y) + cos (2 y))) + C

Solution steps

\int y^{3} cos (2 y) d y

Apply Integration By Parts

: \frac{1}{2} y^{3} sin (2 y) - \int \frac{3}{2} y^{2} sin (2 y) d y

= \frac{1}{2} y^{3} sin (2 y) - \int \frac{3}{2} y^{2} sin (2 y) d y

\int \frac{3}{2} y^{2} sin (2 y) d y = \frac{3}{2} (- \frac{1}{2} y^{2} cos (2 y) + \frac{1}{4} (2 y sin (2 y) + cos (2 y)))

= \frac{1}{2} y^{3} sin (2 y) - \frac{3}{2} (- \frac{1}{2} y^{2} cos (2 y) + \frac{1}{4} (2 y sin (2 y) + cos (2 y)))

Add a constant to the solution

= \frac{1}{2} y^{3} sin (2 y) - \frac{3}{2} (- \frac{1}{2} y^{2} cos (2 y) + \frac{1}{4} (2 y sin (2 y) + cos (2 y))) + C

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What is the integral of y^3cos(2y) ?
The integral of y^3cos(2y) is 1/2 y^3sin(2y)-3/2 (-1/2 y^2cos(2y)+1/4 (2ysin(2y)+cos(2y)))+C