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

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

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

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

\frac{2}{tan ( \frac{x}{2} ) + 1} + x + C

Solution steps

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

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

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

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

\int 1 d x = x

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

Simplify

= \frac{2}{tan ( \frac{x}{2} ) + 1} + x

Add a constant to the solution

= \frac{2}{tan ( \frac{x}{2} ) + 1} + x + C

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