What is the integral of cos(7θ)cos(5θ) ?

The integral of cos(7θ)cos(5θ) is 1/2 (1/12 sin(12θ)+1/2 sin(2θ))+C

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integral of cos(7θ)cos(5θ)

\int cos (7 θ) cos (5 θ) d θ

\frac{1}{2} (\frac{1}{12} sin (12 θ) + \frac{1}{2} sin (2 θ)) + C

Solution steps

\int cos (7 θ) cos (5 θ) d θ

Rewrite using trig identities

= \int \frac{1}{2} (cos (12 θ) + cos (2 θ)) 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 cos (12 θ) + cos (2 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 cos (12 θ) d θ + \int cos (2 θ) d θ)

\int cos (12 θ) d θ = \frac{1}{12} sin (12 θ)

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

= \frac{1}{2} (\frac{1}{12} sin (12 θ) + \frac{1}{2} sin (2 θ))

Add a constant to the solution

= \frac{1}{2} (\frac{1}{12} sin (12 θ) + \frac{1}{2} sin (2 θ)) + C

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What is the integral of cos(7θ)cos(5θ) ?
The integral of cos(7θ)cos(5θ) is 1/2 (1/12 sin(12θ)+1/2 sin(2θ))+C