What is the integral of sin(3w)cos(w) ?

The integral of sin(3w)cos(w) is 1/2 (-1/4 cos(4w)-1/2 cos(2w))+C

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integral of sin(3w)cos(w)

\int sin (3 w) cos (w) d w

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

Solution steps

\int sin (3 w) cos (w) d w

Use the following identity:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

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

Take the constant out:

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int sin (3 w - w) d w = - \frac{1}{2} cos (2 w)

= \frac{1}{2} (- \frac{1}{4} cos (4 w) - \frac{1}{2} cos (2 w))

Add a constant to the solution

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

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What is the integral of sin(3w)cos(w) ?
The integral of sin(3w)cos(w) is 1/2 (-1/4 cos(4w)-1/2 cos(2w))+C