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

The integral of cos(x/2)cos(x/3) is 1/2 (6/5 sin((5x)/6)+6sin(x/6))+C

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

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

\frac{1}{2} (\frac{6}{5} sin (\frac{5 x}{6}) + 6 sin (\frac{x}{6})) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1}{2} (cos (\frac{5 x}{6}) + cos (\frac{x}{6})) 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{5 x}{6}) + cos (\frac{x}{6}) 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{5 x}{6}) d x + \int cos (\frac{x}{6}) d x)

\int cos (\frac{5 x}{6}) d x = \frac{6}{5} sin (\frac{5 x}{6})

\int cos (\frac{x}{6}) d x = 6 sin (\frac{x}{6})

= \frac{1}{2} (\frac{6}{5} sin (\frac{5 x}{6}) + 6 sin (\frac{x}{6}))

Add a constant to the solution

= \frac{1}{2} (\frac{6}{5} sin (\frac{5 x}{6}) + 6 sin (\frac{x}{6})) + C

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What is the integral of cos(x/2)cos(x/3) ?
The integral of cos(x/2)cos(x/3) is 1/2 (6/5 sin((5x)/6)+6sin(x/6))+C