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

The integral of x(sin(2x)) is -1/2 xcos(2x)+1/4 sin(2x)+C

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integral of x(sin(2x))

\int x (sin (2 x)) d x

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

Solution steps

\int x sin (2 x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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