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

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

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integral of cos(2-x/3)

\int cos (2 - \frac{x}{3}) d x

3 sin (\frac{x}{3} - 2) + C

Solution steps

\int cos (2 - \frac{x}{3}) d x

Rewrite using trig identities

= \int cos (2) cos (\frac{x}{3}) + sin (2) sin (\frac{x}{3}) 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) cos (\frac{x}{3}) d x + \int sin (2) sin (\frac{x}{3}) d x

\int cos (2) cos (\frac{x}{3}) d x = cos (2) \cdot 3 sin (\frac{x}{3})

\int sin (2) sin (\frac{x}{3}) d x = - 3 sin (2) cos (\frac{x}{3})

= cos (2) \cdot 3 sin (\frac{x}{3}) - 3 sin (2) cos (\frac{x}{3})

Use the Angle Difference identity:

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

= 3 sin (\frac{x}{3} - 2)

Add a constant to the solution

= 3 sin (\frac{x}{3} - 2) + C

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