What is the integral of (cos(θ/2)-sin((3θ)/2)) ?

The integral of (cos(θ/2)-sin((3θ)/2)) is 2sin(θ/2)+2/3 cos((3θ)/2)+C

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integral of (cos(θ/2)-sin((3θ)/2))

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

2 sin (\frac{θ}{2}) + \frac{2}{3} cos (\frac{3 θ}{2}) + C

Solution steps

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

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 (\frac{θ}{2}) d θ - \int sin (\frac{3 θ}{2}) d θ

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

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

= 2 sin (\frac{θ}{2}) - (- \frac{2}{3} cos (\frac{3 θ}{2}))

Simplify

= 2 sin (\frac{θ}{2}) + \frac{2}{3} cos (\frac{3 θ}{2})

Add a constant to the solution

= 2 sin (\frac{θ}{2}) + \frac{2}{3} cos (\frac{3 θ}{2}) + C

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