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

The integral of (x^2sin(2x)) is -1/2 x^2cos(2x)+1/4 (2xsin(2x)+cos(2x))+C

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

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

- \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x)) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - x cos (2 x) d x = - \frac{1}{4} (2 x sin (2 x) + cos (2 x))

= - \frac{1}{2} x^{2} cos (2 x) - (- \frac{1}{4} (2 x sin (2 x) + cos (2 x)))

Simplify

= - \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x))

Add a constant to the solution

= - \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x)) + C

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What is the integral of (x^2sin(2x)) ?
The integral of (x^2sin(2x)) is -1/2 x^2cos(2x)+1/4 (2xsin(2x)+cos(2x))+C