What is the integral of y^2sin(ny) ?

The integral of y^2sin(ny) is -1/n y^2cos(ny)+2/(n^3)(nysin(ny)+cos(ny))+C

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integral of y^2sin(ny)

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

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

Solution steps

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

Apply Integration By Parts

: - \frac{1}{n} y^{2} cos (n y) - \int - \frac{2}{n} y cos (n y) d y

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

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

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

Simplify

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

Add a constant to the solution

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

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