What is the integral of 5/3 cos(1/3 t)-2sin(2/3 t) ?

The integral of 5/3 cos(1/3 t)-2sin(2/3 t) is 5sin(1/3 t)+3cos((2t)/3)+C

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integral of 5/3 cos(1/3 t)-2sin(2/3 t)

\int \frac{5}{3} cos (\frac{1}{3} t) - 2 sin (\frac{2}{3} t) d t

5 sin (\frac{1}{3} t) + 3 cos (\frac{2 t}{3}) + C

Solution steps

\int \frac{5}{3} cos (\frac{1}{3} t) - 2 sin (\frac{2}{3} t) d t

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{5}{3} cos (\frac{1}{3} t) d t - \int 2 sin (\frac{2}{3} t) d t

\int \frac{5}{3} cos (\frac{1}{3} t) d t = 5 sin (\frac{1}{3} t)

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

= 5 sin (\frac{1}{3} t) - (- 3 cos (\frac{2 t}{3}))

Simplify

= 5 sin (\frac{1}{3} t) + 3 cos (\frac{2 t}{3})

Add a constant to the solution

= 5 sin (\frac{1}{3} t) + 3 cos (\frac{2 t}{3}) + C

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What is the integral of 5/3 cos(1/3 t)-2sin(2/3 t) ?
The integral of 5/3 cos(1/3 t)-2sin(2/3 t) is 5sin(1/3 t)+3cos((2t)/3)+C