What is the integral of x^2(sin(pix)) ?

The integral of x^2(sin(pix)) is -1/pi x^2cos(pix)+2/pi (1/pi xsin(pix)+1/(pi^2)cos(pix))+C

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integral of x^2(sin(pix))

\int x^{2} (sin (π x)) d x

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

Solution steps

\int x^{2} sin (π x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of x^2(sin(pix)) ?
The integral of x^2(sin(pix)) is -1/pi x^2cos(pix)+2/pi (1/pi xsin(pix)+1/(pi^2)cos(pix))+C