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

The integral of sin(2x)x^3 is -1/2 x^3cos(2x)+3/8 (2x^2sin(2x)+2xcos(2x)-sin(2x))+C

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integral of sin(2x)x^3

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

- \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{3}{2} x^{2} cos (2 x) d x = - \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))

= - \frac{1}{2} x^{3} cos (2 x) - (- \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)))

Simplify

= - \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))

Add a constant to the solution

= - \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)) + C

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What is the integral of sin(2x)x^3 ?
The integral of sin(2x)x^3 is -1/2 x^3cos(2x)+3/8 (2x^2sin(2x)+2xcos(2x)-sin(2x))+C