f-tan-x-2f-cot-x-4x-f-x-

Question Number 198562 by cortano12 last updated on 22/Oct/23

f (\tan x) + 2 f (\cot x) = 4 x

f^{'} (x) = ?

Answered by dimentri last updated on 22/Oct/23

l e t x = \frac{π}{2} - t

f (\cot t) + 2 f (\tan t) = 2 π - 4 t

{\begin{cases} f (\cot x) + 2 f (\tan x) = 2 π - 4 x \\ 2 f (\cot t) + f (\tan x) = 4 x \end{cases}

{\begin{cases} 2 f (\cot x) + 4 f (\tan x) = 4 π - 8 x \\ 2 f (\cot x) + f (\tan x) = 4 x \end{cases}

\Leftrightarrow f (\tan x) = \frac{4 π - 12 x}{3}

\Leftrightarrow f (x) = \frac{4 π - 12 \tan^{- 1} x}{3}

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