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

The integral of (x^2*sin(3x)) is -1/3 x^2cos(3x)+2/27 (3xsin(3x)+cos(3x))+C

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integral of (x^2*sin(3x))

\int (x^{2} \cdot sin (3 x)) d x

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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