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

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

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

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

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

Solution steps

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

\frac{sin ( x )}{2 - sin ( x )} = \frac{2}{2 - sin ( x )} - 1

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

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

\int 1 d x = x

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

Add a constant to the solution

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

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