What is the integral of cos(10x+2)cos(8x-2) ?

The integral of cos(10x+2)cos(8x-2) is 1/2 (1/18 sin(18x)+1/2 sin(2x+4))+C

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integral of cos(10x+2)cos(8x-2)

\int cos (10 x + 2) cos (8 x - 2) d x

\frac{1}{2} (\frac{1}{18} sin (18 x) + \frac{1}{2} sin (2 x + 4)) + C

Solution steps

\int cos (10 x + 2) cos (8 x - 2) d x

Rewrite using trig identities

= \int \frac{1}{2} (cos (18 x) + cos (2 x + 4)) 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 (18 x) + cos (2 x + 4) 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 (18 x) d x + \int cos (2 x + 4) d x)

\int cos (18 x) d x = \frac{1}{18} sin (18 x)

\int cos (2 x + 4) d x = \frac{1}{2} sin (2 x + 4)

= \frac{1}{2} (\frac{1}{18} sin (18 x) + \frac{1}{2} sin (2 x + 4))

Add a constant to the solution

= \frac{1}{2} (\frac{1}{18} sin (18 x) + \frac{1}{2} sin (2 x + 4)) + C

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What is the integral of cos(10x+2)cos(8x-2) ?
The integral of cos(10x+2)cos(8x-2) is 1/2 (1/18 sin(18x)+1/2 sin(2x+4))+C