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

The integral of (tan(x)-cot(x))^2 is -4x+tan(x)-cot(x)+C

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

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

- 4 x + tan (x) - cot (x) + C

Solution steps

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

Expand

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

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

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

\int 2 tan (x) cot (x) d x = 2 x

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

= - x + tan (x) - 2 x - x - cot (x)

Simplify

= - 4 x + tan (x) - cot (x)

Add a constant to the solution

= - 4 x + tan (x) - cot (x) + C

s l o p e (x - 3)^{2} + (y - 1)^{2} = 0

d er i v a t i v e y = arcsin (4 x + 1)

a re a y = 2 sin (x), 0 \leq x \leq π

\frac{d y}{d x} = (y - 1)^{2} cos (π x)

d er i v a t i v e g (t) = t^{2} - \frac{4}{t ^{3}}

What is the integral of (tan(x)-cot(x))^2 ?
The integral of (tan(x)-cot(x))^2 is -4x+tan(x)-cot(x)+C