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

The integral of (1-cos(x))/(1+sin(x)) is -2/(tan(x/2)+1)-ln|1+sin(x)|+C

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

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

- \frac{2}{tan ( \frac{x}{2} ) + 1} - ln ∣ 1 + sin (x) ∣ + C

Solution steps

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

Expand

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

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

\int \frac{cos ( x )}{1 + sin ( x )} d x = ln ∣ 1 + sin (x) ∣

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

Add a constant to the solution

= - \frac{2}{tan ( \frac{x}{2} ) + 1} - ln ∣ 1 + sin (x) ∣ + C

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