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

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

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

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

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

Solution steps

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

Long division

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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

\frac{d}{d y} (\frac{x ^{2}}{y})

\int \frac{( 2 - x ) ^{7}}{x} d x

\frac{d}{d y} (e^{x} tan (y))

\frac{d x}{d t} = t - x

\frac{d}{d x} (3 cos^{3} (x))

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