What is the integral of 1/3 cos(1/3 x)-6sin(2/3 x) ?

The integral of 1/3 cos(1/3 x)-6sin(2/3 x) is sin(1/3 x)+9cos((2x)/3)+C

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integral of 1/3 cos(1/3 x)-6sin(2/3 x)

\int \frac{1}{3} cos (\frac{1}{3} x) - 6 sin (\frac{2}{3} x) d x

sin (\frac{1}{3} x) + 9 cos (\frac{2 x}{3}) + C

Solution steps

\int \frac{1}{3} cos (\frac{1}{3} x) - 6 sin (\frac{2}{3} x) 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 \frac{1}{3} cos (\frac{1}{3} x) d x - \int 6 sin (\frac{2}{3} x) d x

\int \frac{1}{3} cos (\frac{1}{3} x) d x = sin (\frac{1}{3} x)

\int 6 sin (\frac{2}{3} x) d x = - 9 cos (\frac{2 x}{3})

= sin (\frac{1}{3} x) - (- 9 cos (\frac{2 x}{3}))

Simplify

= sin (\frac{1}{3} x) + 9 cos (\frac{2 x}{3})

Add a constant to the solution

= sin (\frac{1}{3} x) + 9 cos (\frac{2 x}{3}) + C

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What is the integral of 1/3 cos(1/3 x)-6sin(2/3 x) ?
The integral of 1/3 cos(1/3 x)-6sin(2/3 x) is sin(1/3 x)+9cos((2x)/3)+C