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

Question Number 113628 by mathmax by abdo last updated on 14/Sep/20

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

Answered by MJS_new last updated on 16/Sep/20

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

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

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

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

[t = x + \sqrt{x^{2} - 1} \to d x = \frac{\sqrt{x^{2} - 1}}{x + \sqrt{x^{2} - 1}} d t]

= \frac{1}{8} \int \frac{{(t + 1)}^{2}}{t^{2}} d t = \frac{t^{2} - 1}{8 t} + \frac{1}{4} \ln t =

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

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

[t = x + \sqrt{x^{2} + 1} \to d x = \frac{\sqrt{x^{2} + 1}}{x + \sqrt{x^{2} + 1}}]

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

now we must decompose

t^{4} + t^{3} - t + 1 = (t^{2} + α t + β) (t^{2} + γ t + δ)

α = \frac{1 + \sqrt{5 + 4 \sqrt{2}}}{2}

β = \frac{1 + \sqrt{2}}{2} + \frac{\sqrt{5 + 4 \sqrt{2}} + \sqrt{- 35 + 28 \sqrt{2}}}{8}

γ = \frac{1 - \sqrt{5 + 4 \sqrt{2}}}{2}

δ = \frac{1 + \sqrt{2}}{2} - \frac{\sqrt{5 + 4 \sqrt{2}} + \sqrt{- 35 + 28 \sqrt{2}}}{8}

I^{'} m too tired now

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

[t = x + \sqrt{x^{2} - 1} \to d x = \frac{\sqrt{x^{2} - 1}}{x + \sqrt{x^{2} - 1}}]

= - \frac{1}{8} \int \frac{{(t^{2} - 1)}^{2} (t^{2} + t + 1)}{t^{2} (t^{4} + t^{3} + 4 t^{2} + t + 1)} d t

again to decompose

t^{4} + t^{3} + 4 t^{2} + t + 1 = (t^{2} + a t + b) (t^{2} + c t + d)

a = \frac{1 + \sqrt{- 11 + 8 \sqrt{2}}}{2}

b = \frac{1 + 2 \sqrt{2}}{2} + \frac{\sqrt{- 11 + 8 \sqrt{2}} + \sqrt{77 + 56 \sqrt{2}}}{8}

c = \frac{1 - \sqrt{- 11 + 8 \sqrt{2}}}{2}

d = \frac{1 + 2 \sqrt{2}}{2} - \frac{\sqrt{- 11 + 8 \sqrt{2}} + \sqrt{77 + 56 \sqrt{2}}}{8}

please continue if you need it, I go to bed \dots

Commented by Dwaipayan Shikari last updated on 16/Sep/20

\frac{1}{4} \int \sqrt{\frac{x + 1}{x - 1}}

\frac{1}{4} \int \frac{x + 1}{\sqrt{x^{2} - 1}} = \frac{1}{8} \int \frac{2 x}{\sqrt{x^{2} - 1}} + \frac{1}{4} \int \frac{1}{\sqrt{x^{2} - 1}} = \frac{1}{4} \sqrt{x^{2} - 1} + \frac{1}{4} l o g (x + \sqrt{x^{2} - 1})

Commented by Tawa11 last updated on 06/Sep/21

great sir

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