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

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

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integral of-xcos(2x+3)

\int - x cos (2 x + 3) d x

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

Solution steps

\int - x cos (2 x + 3) d x

Take the constant out:

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

= - \int x cos (2 x + 3) d x

Apply Integration By Parts

: \frac{1}{2} x sin (2 x + 3) - \int \frac{1}{2} sin (2 x + 3) d x

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

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

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

Simplify

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

Add a constant to the solution

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

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