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

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

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integral of 2x^2sin(3x)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \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

= 2 (- \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))

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

Simplify

= 2 (- \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

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