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

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

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

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

- cot (x) - 2 ln ∣ sin (x) ∣ + C

Solution steps

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

Expand

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

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

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

\int 2 cot (x) d x = 2 ln ∣ sin (x) ∣

\int 1 d x = x

= - x - cot (x) - 2 ln ∣ sin (x) ∣ + x

Simplify

= - cot (x) - 2 ln ∣ sin (x) ∣

Add a constant to the solution

= - cot (x) - 2 ln ∣ sin (x) ∣ + C

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