x-2-1-x-2-1-x-2-2x-2-dx-

Question Number 83123 by M±th+et£s last updated on 28/Feb/20

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

Commented by mathmax by abdo last updated on 28/Feb/20

A = \int \frac{x^{2} - 1}{\sqrt{x^{2} + 1} (x^{2} + 2 x - 2)} d x c h a n g e m e n t x = s h (t) g i v e

A = \int \frac{{s h}^{2} t - 1}{c h t ({s h}^{2} t + 2 s h t - 2)} c h t d t

= \int \frac{\frac{c h (2 t) - 1}{2} - 1}{\frac{c h (2 t) - 1}{2} + 2 s h (t) - 2} d t

= \int \frac{c h (2 t) - 3}{c h (2 t) + 4 s h t - 5} d t = \int \frac{\frac{e^{2 t} + e^{- 2 t}}{2} - 3}{\frac{e^{2 t} + e^{- 2 t}}{2} + 2 (e^{t} - e^{- t}) - 5} d t

= \int \frac{e^{2 t} + e^{- 2 t} - 6}{e^{2 t} + e^{- 2 t} + 4 e^{t} - 4 e^{- t} - 10} d t

=_{e^{t} = u} \int \frac{u^{2} + u^{- 2} - 6}{u^{2} + u^{- 2} + 4 u - 4 u^{- 1} - 10} \frac{d u}{u}

= \int \frac{u^{4} + 1 - 6 u^{2}}{u (u^{4} + 1 + 4 u^{3} - 4 u - 10 u^{2})} d u

= \int \frac{u^{4} - 6 u^{2} + 1}{u (u^{4} + 4 u^{3} - 10 u^{2} - 4 u + 1)} d u l e t d e c o m p o s e

F (u) = \frac{u^{4} - 6 u^{2} + 1}{u (u^{4} + 4 u^{3} - 10 u^{2} - 4 u + 1)} \dots . b e c o n t i n u e d \dots

Commented by MJS last updated on 28/Feb/20

u^{4} + 4 u^{3} - 10 u^{2} - 4 u + 1 =

= (u - α) (u - β) (u - γ) (u - δ)

α = - 1 - \sqrt{3} - \sqrt{5 + 2 \sqrt{3}}

β = - 1 - \sqrt{3} + \sqrt{5 + 2 \sqrt{3}}

γ = - 1 + \sqrt{3} - \sqrt{5 - 2 \sqrt{3}}

δ = - 1 + \sqrt{3} + \sqrt{5 - 2 \sqrt{3}}

\dots

Commented by M±th+et£s last updated on 28/Feb/20

t h a n k y o u f o r e v e r y o n e f o r t h e s o l u t i o n

Commented by mathmax by abdo last updated on 28/Feb/20

t h a n k y o u s i r m j s

Commented by MJS last updated on 28/Feb/20

sorry there had been a typo, now corrected

Commented by MJS last updated on 28/Feb/20

{it}^{'} s hard to decompose fractions like this one,

not the method but all these roots \dots

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