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

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

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

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

- x - cot (x) + C

Solution steps

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

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

= \int cot^{2} (x) d x

Rewrite using trig identities

= \int - 1 + csc^{2} (x) 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 1 d x + \int csc^{2} (x) d x

\int 1 d x = x

\int csc^{2} (x) d x = - cot (x)

= - x - cot (x)

Add a constant to the solution

= - x - cot (x) + C

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