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

The integral of ((1-cos(x)))/((1+cos(x))) is 2csc(x)-2cot(x)-x+C

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

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

2 csc (x) - 2 cot (x) - x + C

Solution steps

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

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

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

\int \frac{2 cos ( x )}{1 + cos ( x )} d x = 2 (- csc (x) + cot (x) + x)

\int 1 d x = x

= - 2 (- csc (x) + cot (x) + x) + x

Simplify

- 2 (- csc (x) + cot (x) + x) + x : 2 csc (x) - 2 cot (x) - x

= 2 csc (x) - 2 cot (x) - x

Add a constant to the solution

= 2 csc (x) - 2 cot (x) - x + C

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What is the integral of ((1-cos(x)))/((1+cos(x))) ?
The integral of ((1-cos(x)))/((1+cos(x))) is 2csc(x)-2cot(x)-x+C