What is the integral of y^3sin(y) ?

The integral of y^3sin(y) is -y^3cos(y)+3(y^2sin(y)-2(-ycos(y)+sin(y)))+C

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integral of y^3sin(y)

\int y^{3} sin (y) d y

- y^{3} cos (y) + 3 (y^{2} sin (y) - 2 (- y cos (y) + sin (y))) + C

Solution steps

\int y^{3} sin (y) d y

Apply Integration By Parts

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

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

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

= - y^{3} cos (y) - (- 3 (y^{2} sin (y) - 2 (- y cos (y) + sin (y))))

Simplify

= - y^{3} cos (y) + 3 (y^{2} sin (y) - 2 (- y cos (y) + sin (y)))

Add a constant to the solution

= - y^{3} cos (y) + 3 (y^{2} sin (y) - 2 (- y cos (y) + sin (y))) + C

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What is the integral of y^3sin(y) ?
The integral of y^3sin(y) is -y^3cos(y)+3(y^2sin(y)-2(-ycos(y)+sin(y)))+C