What is the integral of (cos(θ))/(5-3cos(θ)) ?

The integral of (cos(θ))/(5-3cos(θ)) is -1/3 θ+5/6 arctan(2tan(θ/2))+C

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integral of (cos(θ))/(5-3cos(θ))

\int \frac{cos ( θ )}{5 - 3 cos ( θ )} d θ

- \frac{1}{3} θ + \frac{5}{6} arctan (2 tan (\frac{θ}{2})) + C

Solution steps

\int \frac{cos ( θ )}{5 - 3 cos ( θ )} d θ

Long division

\frac{cos ( θ )}{5 - 3 cos ( θ )} : - \frac{1}{3} + \frac{5}{3 ( - 3 cos ( θ ) + 5 )}

= \int - \frac{1}{3} + \frac{5}{3 ( - 3 cos ( θ ) + 5 )} 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 \frac{1}{3} d θ + \int \frac{5}{3 ( - 3 cos ( θ ) + 5 )} d θ

\int \frac{1}{3} d θ = \frac{1}{3} θ

\int \frac{5}{3 ( - 3 cos ( θ ) + 5 )} d θ = \frac{5}{6} arctan (2 tan (\frac{θ}{2}))

= - \frac{1}{3} θ + \frac{5}{6} arctan (2 tan (\frac{θ}{2}))

Add a constant to the solution

= - \frac{1}{3} θ + \frac{5}{6} arctan (2 tan (\frac{θ}{2})) + C

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What is the integral of (cos(θ))/(5-3cos(θ)) ?
The integral of (cos(θ))/(5-3cos(θ)) is -1/3 θ+5/6 arctan(2tan(θ/2))+C