What is the integral of (2x-1)sin(2x) ?

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

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integral of (2x-1)sin(2x)

\int (2 x - 1) sin (2 x) d x

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

Solution steps

\int (2 x - 1) sin (2 x) d x

Expand

(2 x - 1) sin (2 x) : 2 x sin (2 x) - sin (2 x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

Add a constant to the solution

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

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