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

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

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integral of-2xsin(2x)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \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

= - 2 (- \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)

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

Simplify

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

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

Add a constant to the solution

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

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