What is the integral of xsin(2x)cos(2x) ?

The integral of xsin(2x)cos(2x) is -1/8 xcos(4x)+1/32 sin(4x)+C

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integral of xsin(2x)cos(2x)

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

- \frac{1}{8} x cos (4 x) + \frac{1}{32} sin (4 x) + C

Solution steps

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

Apply Integration By Parts

= - \frac{1}{8} x cos (4 x) - \int - \frac{1}{8} cos (4 x) d x

\int - \frac{1}{8} cos (4 x) d x = - \frac{1}{32} sin (4 x)

= - \frac{1}{8} x cos (4 x) - (- \frac{1}{32} sin (4 x))

Simplify

= - \frac{1}{8} x cos (4 x) + \frac{1}{32} sin (4 x)

Add a constant to the solution

= - \frac{1}{8} x cos (4 x) + \frac{1}{32} sin (4 x) + C

What is the integral of xsin(2x)cos(2x) ?
The integral of xsin(2x)cos(2x) is -1/8 xcos(4x)+1/32 sin(4x)+C