What is the integral of sin(3θ)cos(5θ) ?

The integral of sin(3θ)cos(5θ) is 1/2 (-1/8 cos(8θ)+1/2 cos(2θ))+C

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integral of sin(3θ)cos(5θ)

\int sin (3 θ) cos (5 θ) d θ

\frac{1}{2} (- \frac{1}{8} cos (8 θ) + \frac{1}{2} cos (2 θ)) + C

Solution steps

\int sin (3 θ) cos (5 θ) d θ

Use the following identity:

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

= \int \frac{sin ( 3 θ + 5 θ ) + sin ( 3 θ - 5 θ )}{2} d θ

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 θ + 5 θ) + sin (3 θ - 5 θ) d θ

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 θ + 5 θ) d θ + \int sin (3 θ - 5 θ) d θ)

\int sin (3 θ + 5 θ) d θ = - \frac{1}{8} cos (8 θ)

\int sin (3 θ - 5 θ) d θ = \frac{1}{2} cos (2 θ)

= \frac{1}{2} (- \frac{1}{8} cos (8 θ) + \frac{1}{2} cos (2 θ))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{8} cos (8 θ) + \frac{1}{2} cos (2 θ)) + C

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What is the integral of sin(3θ)cos(5θ) ?
The integral of sin(3θ)cos(5θ) is 1/2 (-1/8 cos(8θ)+1/2 cos(2θ))+C