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

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

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integral of (x+1)cos(2x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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