let f(x)=2x−(√(x−1)) find ∫ ((f^(−1) (x))/(f(x)))dx


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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
	$y=2x−(√(x−1)) defined for x≥1 range y≥((17)/(16)) f^(−1) (x)= { ((y=((4x+1−(√(8x−15)))/8); ((15)/8)≤x≤2)),((y=((4x+1+(√(8x−15)))/8); x≥((15)/8))) :} ∫((f^(−1) (x))/(f(x)))dx=(1/8)∫((4x+1)/(2x−(√(x−1))))dx±(1/8)∫((√(8x−15))/(2x−(√(x−1))))dx it′s easy to solve the first one but the second one takes a lot of time and space...$
	$y = 2 x - \sqrt{x - 1}$ $defined for x ⩾ 1$ $range y ⩾ \frac{17}{16}$ $f^{- 1} (x) = {\begin{cases} y = \frac{4 x + 1 - \sqrt{8 x - 15}}{8}; \frac{15}{8} ⩽ x ⩽ 2 \\ y = \frac{4 x + 1 + \sqrt{8 x - 15}}{8}; x ⩾ \frac{15}{8} \end{cases}$ $\int \frac{f^{- 1} (x)}{f (x)} d x = \frac{1}{8} \int \frac{4 x + 1}{2 x - \sqrt{x - 1}} d x \pm \frac{1}{8} \int \frac{\sqrt{8 x - 15}}{2 x - \sqrt{x - 1}} d x$ ${it}^{'} s easy to solve the first one but the second$ $one takes a lot of time and space . . .$
	Commented by mathmax by abdo last updated on 19/Apr/20

	$t h a n k y o u s i r$
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