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

The integral of cos(x-2y) is sin(x-2y)+C

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integral of cos(x-2y)

\int cos (x - 2 y) d x

sin (x - 2 y) + C

Solution steps

\int cos (x - 2 y) d x

Rewrite using trig identities

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

\int cos (x) cos (2 y) d x = cos (2 y) sin (x)

\int sin (x) sin (2 y) d x = - sin (2 y) cos (x)

= cos (2 y) sin (x) - sin (2 y) cos (x)

Use the Angle Difference identity:

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

= sin (x - 2 y)

Add a constant to the solution

= sin (x - 2 y) + C