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

The integral of sin(2x)cos(x/3) is 1/2 (-3/7 cos((7x)/3)-3/5 cos((5x)/3))+C

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

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

\frac{1}{2} (- \frac{3}{7} cos (\frac{7 x}{3}) - \frac{3}{5} cos (\frac{5 x}{3})) + C

Solution steps

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

Use the following identity:

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

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

Take the constant out:

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

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

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

\int sin (2 x + \frac{x}{3}) d x = - \frac{3}{7} cos (\frac{7 x}{3})

\int sin (2 x - \frac{x}{3}) d x = - \frac{3}{5} cos (\frac{5 x}{3})

= \frac{1}{2} (- \frac{3}{7} cos (\frac{7 x}{3}) - \frac{3}{5} cos (\frac{5 x}{3}))

Add a constant to the solution

= \frac{1}{2} (- \frac{3}{7} cos (\frac{7 x}{3}) - \frac{3}{5} cos (\frac{5 x}{3})) + C

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What is the integral of sin(2x)cos(x/3) ?
The integral of sin(2x)cos(x/3) is 1/2 (-3/7 cos((7x)/3)-3/5 cos((5x)/3))+C