What is the integral of cos(x/5)cos(x/3) ?

The integral of cos(x/5)cos(x/3) is 1/2 (15/2 sin((2x)/(15))+15/8 sin((8x)/(15)))+C

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integral of cos(x/5)cos(x/3)

\int cos (\frac{x}{5}) cos (\frac{x}{3}) d x

\frac{1}{2} (\frac{15}{2} sin (\frac{2 x}{15}) + \frac{15}{8} sin (\frac{8 x}{15})) + C

Solution steps

\int cos (\frac{x}{5}) cos (\frac{x}{3}) d x

Rewrite using trig identities

= \int \frac{1}{2} (cos (\frac{8 x}{15}) + cos (- \frac{2 x}{15})) 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 (\frac{8 x}{15}) + cos (- \frac{2 x}{15}) d x

Simplify

cos (\frac{8 x}{15}) + cos (- \frac{2 x}{15}) : cos (\frac{2 x}{15}) + cos (\frac{8 x}{15})

= \frac{1}{2} \cdot \int cos (\frac{2 x}{15}) + cos (\frac{8 x}{15}) 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 (\frac{2 x}{15}) d x + \int cos (\frac{8 x}{15}) d x)

\int cos (\frac{2 x}{15}) d x = \frac{15}{2} sin (\frac{2 x}{15})

\int cos (\frac{8 x}{15}) d x = \frac{15}{8} sin (\frac{8 x}{15})

= \frac{1}{2} (\frac{15}{2} sin (\frac{2 x}{15}) + \frac{15}{8} sin (\frac{8 x}{15}))

Add a constant to the solution

= \frac{1}{2} (\frac{15}{2} sin (\frac{2 x}{15}) + \frac{15}{8} sin (\frac{8 x}{15})) + C

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What is the integral of cos(x/5)cos(x/3) ?
The integral of cos(x/5)cos(x/3) is 1/2 (15/2 sin((2x)/(15))+15/8 sin((8x)/(15)))+C