What is the integral of (sin(3x)+cos(5x)-sin(x/2)) ?

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

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integral of (sin(3x)+cos(5x)-sin(x/2))

\int (sin (3 x) + cos (5 x) - sin (\frac{x}{2})) d x

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

Solution steps

\int sin (3 x) + cos (5 x) - sin (\frac{x}{2}) 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 sin (3 x) d x + \int cos (5 x) d x - \int sin (\frac{x}{2}) d x

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

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

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

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

Simplify

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

Add a constant to the solution

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

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