let-f-x-2x-x-1-find-f-1-x-f-x-dx-

Question Number 89717 by abdomathmax last updated on 18/Apr/20

l e t f (x) = 2 x - \sqrt{x - 1}

f i n d \int \frac{f^{- 1} (x)}{f (x)} d x

Answered by MJS last updated on 19/Apr/20

y = 2 x - \sqrt{x - 1}

defined for x ⩾ 1

range y ⩾ \frac{17}{16}

f^{- 1} (x) = {\begin{cases} y = \frac{4 x + 1 - \sqrt{8 x - 15}}{8}; \frac{15}{8} ⩽ x ⩽ 2 \\ y = \frac{4 x + 1 + \sqrt{8 x - 15}}{8}; x ⩾ \frac{15}{8} \end{cases}

\int \frac{f^{- 1} (x)}{f (x)} d x = \frac{1}{8} \int \frac{4 x + 1}{2 x - \sqrt{x - 1}} d x \pm \frac{1}{8} \int \frac{\sqrt{8 x - 15}}{2 x - \sqrt{x - 1}} d x

{it}^{'} s easy to solve the first one but the second

one takes a lot of time and space \dots

Commented by mathmax by abdo last updated on 19/Apr/20

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