What is the integral of (1-cot(2x))^2 ?

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

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integral of (1-cot(2x))^2

\int (1 - cot (2 x))^{2} d x

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

Solution steps

\int (1 - cot (2 x))^{2} d x

Expand

(1 - cot (2 x))^{2} : 1 - 2 cot (2 x) + cot^{2} (2 x)

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

\int 1 d x = x

\int 2 cot (2 x) d x = ln ∣ sin (2 x) ∣

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

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

Simplify

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

Add a constant to the solution

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

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