calculate-0-dx-x-2-1-x-2-

Question Number 37961 by prof Abdo imad last updated on 19/Jun/18

c a l c u l a t e \int_{0}^{\infty} \frac{d x}{x^{2} + \sqrt{1 + x^{2}}} .

Answered by MJS last updated on 20/Jun/18

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

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

= N_{1} (\int \frac{d x}{x - \frac{\sqrt{2}}{2} \sqrt{1 + \sqrt{5}}} - \int \frac{d x}{x + \frac{\sqrt{2}}{2} \sqrt{1 + \sqrt{5}}}) + N_{2} \int \frac{d x}{x^{2} - \frac{1}{2} + \frac{\sqrt{5}}{2}} =

= \frac{\sqrt{10}}{20} \sqrt{1 + \sqrt{5}} \ln ∣ \frac{x - \frac{\sqrt{2}}{2} \sqrt{1 + \sqrt{5}}}{x + \frac{\sqrt{2}}{2} \sqrt{1 + \sqrt{5}}} ∣ + \frac{\sqrt{10}}{10} \sqrt{- 1 + \sqrt{5}} \arctan (\frac{\sqrt{2}}{2} \sqrt{1 + \sqrt{5}} x)

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

[t = \frac{x}{\sqrt{x^{2} + 1}} \to d x = \sqrt{{(x^{2} + 1)}^{3}} d t]

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

= N_{3} (\int \frac{d t}{t - \frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}}} - \int \frac{d t}{t + \frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}}}) + N_{4} \int \frac{d t}{t^{2} + \frac{1}{2} + \frac{\sqrt{5}}{2}} =

= \frac{\sqrt{10}}{20} \sqrt{1 + \sqrt{5}} \ln ∣ \frac{t - \frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}}}{t + \frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}}} ∣ - \frac{\sqrt{10}}{10} \sqrt{- 1 + \sqrt{5}} \arctan (\frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}} t) =

= \frac{\sqrt{10}}{20} \sqrt{1 + \sqrt{5}} \ln ∣ \frac{\frac{x}{\sqrt{x^{2} + 1}} - \frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}}}{\frac{x}{\sqrt{x^{2} + 1}} + \frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}}} ∣ - \frac{\sqrt{10}}{10} \sqrt{- 1 + \sqrt{5}} \arctan (\frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}} \frac{x}{\sqrt{x^{2} + 1}})

= \frac{\sqrt{10}}{20} (\sqrt{1 + \sqrt{5}} (\ln ∣ \frac{x - \frac{\sqrt{2}}{2} \sqrt{1 + \sqrt{5}}}{x + \frac{\sqrt{2}}{2} \sqrt{1 + \sqrt{5}}} ∣ - \ln ∣ \frac{\frac{x}{\sqrt{x^{2} + 1}} - \frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}}}{\frac{x}{\sqrt{x^{2} + 1}} + \frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}}} ∣) + 2 \sqrt{- 1 + \sqrt{5}} (\arctan (\frac{\sqrt{2}}{2} \sqrt{1 + \sqrt{5}} x) + \arctan (\frac{\sqrt{2}}{2} \sqrt{- 1 + \sqrt{5}} \frac{x}{\sqrt{x^{2} + 1}}))) + C

\underset{0}{\int^{\infty}} \frac{d x}{x^{2} + \sqrt{x^{2} + 1}} = \frac{\sqrt{10}}{20} (\sqrt{1 + \sqrt{5}} \ln (2 + \sqrt{5} + 2 \sqrt{2 + \sqrt{5}}) + \sqrt{- 1 + \sqrt{5}} (π + 2 \arcsin (\frac{- 1 + \sqrt{5}}{2}))) \approx 1.39021

Commented by math khazana by abdo last updated on 20/Jun/18

t h a n k y o u s i r f o r t h i s h a r d w o r k .