What is the integral of xsin(2x+3) ?

The integral of xsin(2x+3) is -1/2 xcos(2x+3)+1/4 sin(2x+3)+C

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integral of xsin(2x+3)

\int x sin (2 x + 3) d x

- \frac{1}{2} x cos (2 x + 3) + \frac{1}{4} sin (2 x + 3) + C

Solution steps

\int x sin (2 x + 3) d x

Apply Integration By Parts

= - \frac{1}{2} x cos (2 x + 3) - \int - \frac{1}{2} cos (2 x + 3) d x

\int - \frac{1}{2} cos (2 x + 3) d x = - \frac{1}{4} sin (2 x + 3)

= - \frac{1}{2} x cos (2 x + 3) - (- \frac{1}{4} sin (2 x + 3))

Simplify

= - \frac{1}{2} x cos (2 x + 3) + \frac{1}{4} sin (2 x + 3)

Add a constant to the solution

= - \frac{1}{2} x cos (2 x + 3) + \frac{1}{4} sin (2 x + 3) + C

What is the integral of xsin(2x+3) ?
The integral of xsin(2x+3) is -1/2 xcos(2x+3)+1/4 sin(2x+3)+C