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

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

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integral of (sin(θ))/(1+sin(θ))

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

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

Solution steps

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

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

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

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 ( θ )} d θ + \int 1 d θ

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

\int 1 d θ = θ

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

Simplify

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

Add a constant to the solution

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

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