cos-2cot-1-1-x-1-x-dx-

Question Number 118575 by bramlexs22 last updated on 18/Oct/20

\int \cos ({2 \cot}^{- 1} \sqrt{\frac{1 - x}{1 + x}}) d x ?

Answered by benjo_mathlover last updated on 18/Oct/20

s e t \cot^{- 1} (\sqrt{\frac{1 - x}{1 + x}}) = ϕ \Rightarrow \sqrt{\frac{1 - x}{1 + x}} = \cot ϕ

\Rightarrow \cot^{2} ϕ = \frac{1 - x}{1 + x}, \tan^{2} ϕ = \frac{1 + x}{1 - x}

\Rightarrow \sec^{2} ϕ = 1 + \frac{1 + x}{1 - x} = \frac{2}{1 - x}

\Rightarrow \cos^{2} ϕ = \frac{1 - x}{2} \land \cos 2 ϕ = 2 \cos^{2} ϕ - 1

\Rightarrow \cos 2 ϕ = 2 (\frac{1 - x}{2}) - 1 = - x

T h u s \int \cos ({2 \cot}^{- 1} (\sqrt{\frac{1 - x}{1 + x}})) d x =

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