What is the integral of cos(2x-5) ?

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

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integral of cos(2x-5)

\int cos (2 x - 5) d x

cos (5) \frac{1}{2} sin (2 x) - \frac{1}{2} sin (5) cos (2 x) + C

Solution steps

\int cos (2 x - 5) d x

Rewrite using trig identities

= \int cos (2 x) cos (5) + sin (2 x) sin (5) 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 (5) d x + \int sin (2 x) sin (5) d x

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

\int sin (2 x) sin (5) d x = - \frac{1}{2} sin (5) cos (2 x)

= cos (5) \frac{1}{2} sin (2 x) - \frac{1}{2} sin (5) cos (2 x)

Add a constant to the solution

= cos (5) \frac{1}{2} sin (2 x) - \frac{1}{2} sin (5) cos (2 x) + C

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