2-x-2-dx-

Question Number 146678 by mathdanisur last updated on 14/Jul/21

\int \sqrt{2 + x^{2}} d x = ?

Answered by Ar Brandon last updated on 14/Jul/21

I = \int \sqrt{2 + x^{2}} dx

= 2 \int \sqrt{1 + \sinh^{2} θ} \cdot \cosh θ d θ

= 2 \int \cosh^{2} θ = \int (\cosh 2 θ + 1) d θ

= \frac{\sinh 2 θ}{2} + θ = \sinh θ \cosh θ + θ + C

= \frac{x}{2} \sqrt{2 + x^{2}} + argsinh (\frac{x}{2}) + C

Commented by mathdanisur last updated on 15/Jul/21

t h a n k y o u S e r c o o l

Answered by mathmax by abdo last updated on 14/Jul/21

I = \int \sqrt{2 + x^{2}} dx changement x = \sqrt{2} sht give

I = \int \sqrt{2} cht (\sqrt{2}) cht dt = \int {2 ch}^{2} t dt = \int (1 + ch (2 t)) dt

= t + \frac{1}{2} sh (2 t) = t + sht cht but t = argsh (\frac{x}{\sqrt{2}}) = \log (\frac{x}{\sqrt{2}} + \sqrt{1 + \frac{x^{2}}{2}})

\Rightarrow I = \log (\frac{x}{\sqrt{2}} + \sqrt{1 + \frac{x^{2}}{2}}) + \frac{x}{\sqrt{2}} \sqrt{1 + \frac{x^{2}}{2}} + c

= \log (x + \sqrt{2 + x^{2}}) + \frac{x}{2} \sqrt{2 + x^{2}} + C

Commented by mathdanisur last updated on 15/Jul/21

t h a n k y o u S e r c o o l