What is the integral of (sin(2x)*3x)/2 ?

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

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integral of (sin(2x)*3x)/2

\int \frac{sin ( 2 x ) \cdot 3 x}{2} d x

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

Solution steps

\int \frac{sin ( 2 x ) \cdot 3 x}{2} d x

Take the constant out:

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

= \frac{3}{2} \cdot \int sin (2 x) x d x

Apply Integration By Parts

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

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

Simplify

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

Add a constant to the solution

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

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