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

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

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

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

- cot (\frac{x}{2}) - x + C

Solution steps

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

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

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

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

\int 1 d x = x

= - cot (\frac{x}{2}) - x

Add a constant to the solution

= - cot (\frac{x}{2}) - x + C

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