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

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

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integral of u^2sin(2u)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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