What is the integral of t^2sin(3t) ?

The integral of t^2sin(3t) is -1/3 t^2cos(3t)+2/27 (3tsin(3t)+cos(3t))+C

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integral of t^2sin(3t)

\int t^{2} sin (3 t) d t

- \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t)) + C

Solution steps

\int t^{2} sin (3 t) d t

Apply Integration By Parts

: - \frac{1}{3} t^{2} cos (3 t) - \int - \frac{2}{3} t cos (3 t) d t

= - \frac{1}{3} t^{2} cos (3 t) - \int - \frac{2}{3} t cos (3 t) d t

\int - \frac{2}{3} t cos (3 t) d t = - \frac{2}{27} (3 t sin (3 t) + cos (3 t))

= - \frac{1}{3} t^{2} cos (3 t) - (- \frac{2}{27} (3 t sin (3 t) + cos (3 t)))

Simplify

= - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t))

Add a constant to the solution

= - \frac{1}{3} t^{2} cos (3 t) + \frac{2}{27} (3 t sin (3 t) + cos (3 t)) + C

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What is the integral of t^2sin(3t) ?
The integral of t^2sin(3t) is -1/3 t^2cos(3t)+2/27 (3tsin(3t)+cos(3t))+C