2-2-dx-x-2-x-2-1-3-

Question Number 138570 by bemath last updated on 15/Apr/21

\underset{\sqrt{2}}{\int^{2}} \frac{d x}{x^{2} \sqrt{{(x^{2} - 1)}^{3}}} = ?

Answered by Ar Brandon last updated on 15/Apr/21

I = \int_{\sqrt{2}}^{2} \frac{dx}{x^{2} \sqrt{{(x^{2} - 1)}^{3}}} = \int_{\sqrt{2}}^{2} \frac{dx}{x^{5} \sqrt{{(1 - \frac{1}{x^{2}})}^{3}}}, u = \frac{1}{x^{2}}

- \frac{1}{2} \int \frac{udu}{\sqrt{{(1 - u)}^{3}}} = - \frac{u}{\sqrt{(1 - u)}} + \int \frac{du}{\sqrt{1 - u}}

= - \frac{u}{\sqrt{1 - u}} - 2 \sqrt{1 - u} + C