What is the integral of (sin(2x))/(1+sin(2x)) ?

The integral of (sin(2x))/(1+sin(2x)) is x+1/(tan(x)+1)+C

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integral of (sin(2x))/(1+sin(2x))

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

x + \frac{1}{tan ( x ) + 1} + C

Solution steps

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

Long division

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

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

\int 1 d x = x

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

= x - (- \frac{1}{tan ( x ) + 1})

Simplify

= x + \frac{1}{tan ( x ) + 1}

Add a constant to the solution

= x + \frac{1}{tan ( x ) + 1} + C

What is the integral of (sin(2x))/(1+sin(2x)) ?
The integral of (sin(2x))/(1+sin(2x)) is x+1/(tan(x)+1)+C