What is the integral of cos(5x)cos(10x) ?

The integral of cos(5x)cos(10x) is 1/2 (1/5 sin(5x)+1/15 sin(15x))+C

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integral of cos(5x)cos(10x)

\int cos (5 x) cos (10 x) d x

\frac{1}{2} (\frac{1}{5} sin (5 x) + \frac{1}{15} sin (15 x)) + C

Solution steps

\int cos (5 x) cos (10 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (cos (15 x) + cos (- 5 x)) 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 cos (15 x) + cos (- 5 x) d x

Simplify

cos (15 x) + cos (- 5 x) : cos (5 x) + cos (15 x)

= \frac{1}{2} \cdot \int cos (5 x) + cos (15 x) 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 cos (5 x) d x + \int cos (15 x) d x)

\int cos (5 x) d x = \frac{1}{5} sin (5 x)

\int cos (15 x) d x = \frac{1}{15} sin (15 x)

= \frac{1}{2} (\frac{1}{5} sin (5 x) + \frac{1}{15} sin (15 x))

Add a constant to the solution

= \frac{1}{2} (\frac{1}{5} sin (5 x) + \frac{1}{15} sin (15 x)) + C

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What is the integral of cos(5x)cos(10x) ?
The integral of cos(5x)cos(10x) is 1/2 (1/5 sin(5x)+1/15 sin(15x))+C