2-x-2-2-x-2-4-x-4-dx-

Question Number 147749 by cesarL last updated on 23/Jul/21

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

Answered by mathmax by abdo last updated on 23/Jul/21

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

= \int \frac{dx}{\sqrt{2 + x^{2}}} dx + \int \frac{dx}{\sqrt{2 - x^{2}}} dx we have

\int \frac{dx}{\sqrt{2 + x^{2}}} dx =_{x = \sqrt{2} t} \int \frac{\sqrt{2} dt}{\sqrt{2} . \sqrt{1 + t^{2}}} dt = \log (t + \sqrt{1 + t^{2}}) + c_{0}

= \log (\frac{x}{\sqrt{2}} + \sqrt{1 + \frac{x^{2}}{2}}) + c_{0} = \log (x + \sqrt{2 + x^{2}}) + k_{0}

\int \frac{dx}{\sqrt{2 - x^{2}}} =_{x = \sqrt{2} t} \int \frac{\sqrt{2} dt}{\sqrt{2} \sqrt{1 - t^{2}}} = arcsint + c_{1} = \arcsin (\frac{x}{\sqrt{2}}) + c_{1} \Rightarrow

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