f-x-x-2-2x-5-find-f-x-f-1-x-dx-and-f-1-x-f-x-dx-

Question Number 147466 by mathmax by abdo last updated on 21/Jul/21

f (x) = x^{2} - 2 x + 5

find \int \frac{f (x)}{f^{- 1} (x)} dx and \int \frac{f^{- 1} (x)}{f (x)} dx

Answered by EDWIN88 last updated on 21/Jul/21

f (x) = {(x - 1)}^{2} + 4 \Rightarrow x = \pm \sqrt{x - 4} + 1

f^{- 1} (x) = 1 \pm \sqrt{x - 4}

(1) \int \frac{{(x - 1)}^{2} + 4}{1 + \sqrt{x - 4}} dx =

let x - 4 = u^{2} \to dx = 2 u du

\int \frac{{(u^{2} + 3)}^{2} + 4}{1 + u} (2 u du) =

2 \int \frac{u^{4} + {6 u}^{3} + 13 u}{u + 1} =

2 \int (u^{3} + {5 u}^{2} - 5 u + 18 - \frac{18}{u + 1}) du =

2 (\frac{1}{4} u^{4} + \frac{5}{3} u^{3} - \frac{5}{2} u^{2} + 18 u - 18 \ln (u + 1)) + c =

\frac{1}{2} {(x - 4)}^{2} + \frac{10}{3} {(x - 4)}^{3 / 2} - 5 (x - 4) + 36 \sqrt{x - 4} - 36 \ln (1 + \sqrt{x - 4}) + c

Answered by mathmax by abdo last updated on 21/Jul/21

f (x) = y \Rightarrow x = f^{- 1} (y) \Rightarrow x^{2} - 2 x + 5 = y \Rightarrow x^{2} - 2 x + 5 - y = 0

Δ^{^{'}} = 1 - (5 - y) = y - 4 for y ⩾ 4 we get

x_{1} = 1 + \sqrt{y - 4} and x_{2} = 1 - \sqrt{y - 4}

\Rightarrow f^{- 1} (x) = 1 + \sqrt{x - 4} or f^{- 1} (x) = 1 - \sqrt{x - 4}

case 1 \int \frac{f (x)}{f^{- 1} (x)} dx = \int \frac{x^{2} - 2 x + 5}{1 + \sqrt{x - 4}} dx =_{x - 4 = y^{2}} \int \frac{{(4 + y^{2})}^{2} - 2 (4 + y^{2}) + 5}{1 + y} 2 ydy

= 2 \int \frac{16 + {8 y}^{2} + y^{4} - 8 - {2 y}^{2} + 5}{y + 1} ydy

= 2 \int \frac{y^{5} + {6 y}^{3} + 13 y}{y + 1} dy =_{y + 1 = t} 2 \int \frac{{(t - 1)}^{5} + 6 {(t - 1)}^{3} + 13 (t - 1)}{t} dt

= 2 \int \frac{\sum_{k = 0}^{5} C_{5}^{k} t^{k} {(- 1)}^{5 - k} + 6 (t^{3} - {3 t}^{2} + 3 t - 1) + 13 t - 13}{t} dt

= - 2 \sum_{k = 0}^{5} C_{5}^{k} {(- 1)}^{k} \int t^{k - 1} dt + 12 \int (t^{2} - 3 t + 3 - \frac{1}{t}) dt + 2 \int (13 - \frac{13}{t} dt

= - 2 \log ∣ t ∣ - 2 \sum_{k = 1}^{5} C_{5}^{k} {(- 1)}^{k} \frac{t^{k}}{k} + 12 {\frac{t^{3}}{3} - \frac{3}{2} t^{2} + 3 t - \log ∣ t ∣)

+ 26 t - 26 \log ∣ t ∣ + C

t = y + 1 = \sqrt{x - 4} + 1 \Rightarrow

\int \frac{f (x)}{f^{- 1} (x)} dx = - 2 \log ∣ \sqrt{x - 4} + 1 ∣ - 2 \sum_{k = 1}^{5} C_{5}^{k} \frac{{(- 1)}^{k}}{k} {(\sqrt{x - 4} + 1)}^{k}

+ 4 {(1 + \sqrt{x - 4})}^{3} - 18 {(1 + \sqrt{x - 4})}^{2} + 36 (1 + \sqrt{x - 4}) + 3 (1 + \sqrt{x - 4})

- \log ∣ 1 + \sqrt{x - 4} ∣ + 26 (1 + \sqrt{x - 4}) - 26 \log ∣ 1 + \sqrt{x - 4} ∣ + C

rest to treat case 2 \dots . . be continued \dots

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