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

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

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

\int cos (3 x) cos (2 x) 2 d x

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

Solution steps

\int cos (3 x) cos (2 x) \cdot 2 d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int cos (3 x) cos (2 x) d x

Rewrite using trig identities

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

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

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

\int cos (x) d x = sin (x)

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

Simplify

2 \cdot \frac{1}{2} (\frac{1}{5} sin (5 x) + sin (x)) : \frac{1}{5} sin (5 x) + sin (x)

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

Add a constant to the solution

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

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