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

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

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

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

\frac{1}{3} sin (\frac{3 x}{2}) + sin (\frac{x}{2}) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1}{2} (cos (\frac{3 x}{2}) + cos (\frac{x}{2})) 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{3 x}{2}) + cos (\frac{x}{2}) 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{3 x}{2}) d x + \int cos (\frac{x}{2}) d x)

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

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

= \frac{1}{2} (\frac{2}{3} sin (\frac{3 x}{2}) + 2 sin (\frac{x}{2}))

Simplify

\frac{1}{2} (\frac{2}{3} sin (\frac{3 x}{2}) + 2 sin (\frac{x}{2})) : \frac{1}{3} sin (\frac{3 x}{2}) + sin (\frac{x}{2})

= \frac{1}{3} sin (\frac{3 x}{2}) + sin (\frac{x}{2})

Add a constant to the solution

= \frac{1}{3} sin (\frac{3 x}{2}) + sin (\frac{x}{2}) + C

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