What is the integral of (1-cos(x))/(1-cos(2x)) ?

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

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integral of (1-cos(x))/(1-cos(2x))

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

- \frac{1}{2} cot (x) + \frac{1}{2} csc (x) + C

Solution steps

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

Expand

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

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

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

= - \frac{1}{2} cot (x) - (- \frac{1}{2} csc (x))

Simplify

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

Add a constant to the solution

= - \frac{1}{2} cot (x) + \frac{1}{2} csc (x) + C

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