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

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

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integral of xsin(2x-1)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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