What is the integral of ysin(2y) ?

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

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integral of ysin(2y)

\int y sin (2 y) d y

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

Solution steps

\int y sin (2 y) d y

Apply Integration By Parts

= - \frac{1}{2} y cos (2 y) - \int - \frac{1}{2} cos (2 y) d y

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

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

Simplify

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

Add a constant to the solution

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

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