What is the integral of x^2sin(5x) ?

The integral of x^2sin(5x) is -1/5 x^2cos(5x)+2/125 (5xsin(5x)+cos(5x))+C

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integral of x^2sin(5x)

\int x^{2} sin (5 x) d x

- \frac{1}{5} x^{2} cos (5 x) + \frac{2}{125} (5 x sin (5 x) + cos (5 x)) + C

Solution steps

\int x^{2} sin (5 x) d x

Apply Integration By Parts

= - \frac{1}{5} x^{2} cos (5 x) - \int - \frac{2}{5} x cos (5 x) d x

\int - \frac{2}{5} x cos (5 x) d x = - \frac{2}{125} (5 x sin (5 x) + cos (5 x))

= - \frac{1}{5} x^{2} cos (5 x) - (- \frac{2}{125} (5 x sin (5 x) + cos (5 x)))

Simplify

= - \frac{1}{5} x^{2} cos (5 x) + \frac{2}{125} (5 x sin (5 x) + cos (5 x))

Add a constant to the solution

= - \frac{1}{5} x^{2} cos (5 x) + \frac{2}{125} (5 x sin (5 x) + cos (5 x)) + C

What is the integral of x^2sin(5x) ?
The integral of x^2sin(5x) is -1/5 x^2cos(5x)+2/125 (5xsin(5x)+cos(5x))+C