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

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

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integral of ysin(y)cos^2(y)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

- \frac{1}{3} y cos^{3} (y) - (- \frac{1}{3} (sin (y) - \frac{sin ^{3} ( y )}{3})) : - \frac{1}{3} y cos^{3} (y) + \frac{1}{3} (sin (y) - \frac{1}{3} sin^{3} (y))

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

Add a constant to the solution

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

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