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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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