What is the integral of cos(2x-6) ?

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

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integral of cos(2x-6)

\int cos (2 x - 6) d x

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

Solution steps

\int cos (2 x - 6) d x

Rewrite using trig identities

= \int cos (2 x) cos (6) + sin (2 x) sin (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

= \int cos (2 x) cos (6) d x + \int sin (2 x) sin (6) d x

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

\int sin (2 x) sin (6) d x = - \frac{1}{2} sin (6) cos (2 x)

= cos (6) \frac{1}{2} sin (2 x) - \frac{1}{2} sin (6) cos (2 x)

Use the Angle Difference identity:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

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

Add a constant to the solution

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

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