What is the integral of 2/3 cos(1/3)x-8sin(2/3)x ?

The integral of 2/3 cos(1/3)x-8sin(2/3)x is 1/3 cos(1/3)x^2-4sin(2/3)x^2+C

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integral of 2/3 cos(1/3)x-8sin(2/3)x

\int \frac{2}{3} cos (\frac{1}{3}) x - 8 sin (\frac{2}{3}) x d x

\frac{1}{3} cos (\frac{1}{3}) x^{2} - 4 sin (\frac{2}{3}) x^{2} + C

Solution steps

\int \frac{2}{3} cos (\frac{1}{3}) x - 8 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{2}{3} cos (\frac{1}{3}) x d x - \int 8 sin (\frac{2}{3}) x d x

\int \frac{2}{3} cos (\frac{1}{3}) x d x = \frac{1}{3} cos (\frac{1}{3}) x^{2}

\int 8 sin (\frac{2}{3}) x d x = 4 sin (\frac{2}{3}) x^{2}

= \frac{1}{3} cos (\frac{1}{3}) x^{2} - 4 sin (\frac{2}{3}) x^{2}

Add a constant to the solution

= \frac{1}{3} cos (\frac{1}{3}) x^{2} - 4 sin (\frac{2}{3}) x^{2} + C

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What is the integral of 2/3 cos(1/3)x-8sin(2/3)x ?
The integral of 2/3 cos(1/3)x-8sin(2/3)x is 1/3 cos(1/3)x^2-4sin(2/3)x^2+C