What is the integral of xcos(2x+1) ?

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

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integral of xcos(2x+1)

\int x cos (2 x + 1) d x

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

Solution steps

\int x cos (2 x + 1) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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