What is the integral of (cos(3x)-sin(5x)) ?

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

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integral of (cos(3x)-sin(5x))

\int (cos (3 x) - sin (5 x)) d x

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

Solution steps

\int cos (3 x) - sin (5 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 cos (3 x) d x - \int sin (5 x) d x

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

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

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

Simplify

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

Add a constant to the solution

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

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