What is the integral of (cos(x))/(2+cos(x)) ?

The integral of (cos(x))/(2+cos(x)) is -4/(sqrt(3))arctan(1/(sqrt(3))tan(x/2))+x+C

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integral of (cos(x))/(2+cos(x))

\int \frac{cos ( x )}{2 + cos ( x )} d x

- \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + x + C

Solution steps

\int \frac{cos ( x )}{2 + cos ( x )} d x

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

= \int - \frac{2}{2 + cos ( x )} + 1 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{2}{2 + cos ( x )} d x + \int 1 d x

\int \frac{2}{2 + cos ( x )} d x = \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2}))

\int 1 d x = x

= - \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + x

Add a constant to the solution

= - \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + x + C

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What is the integral of (cos(x))/(2+cos(x)) ?
The integral of (cos(x))/(2+cos(x)) is -4/(sqrt(3))arctan(1/(sqrt(3))tan(x/2))+x+C