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

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

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

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

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

Solution steps

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

Long division

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

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

\int \frac{1}{3} d x = \frac{1}{3} x

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

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

Add a constant to the solution

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

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