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

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

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integral of t^2sin(pit)

\int t^{2} sin (π t) d t

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

Solution steps

\int t^{2} sin (π t) d t

Apply Integration By Parts

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

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

\int - \frac{2}{π} t cos (π t) d t = - \frac{2}{π} (\frac{1}{π} t sin (π t) + \frac{1}{π ^{2}} cos (π t))

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

Simplify

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

Add a constant to the solution

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

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