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

Question Number 206096 by RoseAli last updated on 06/Apr/24

\int \frac{1}{x^{3} \sqrt{x^{2} - 1}} . d x

Answered by Frix last updated on 07/Apr/24

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

From here we could use {Ostrogradski}^{'} s

Method or this :

\int \frac{d t}{{(t^{2} + 1)}^{2}} \overset{u = \tan^{- 1} t}{=} \int \cos^{2} u d u =

= \frac{1}{2} \int (1 + \cos 2 u) d u =

= \frac{u}{2} + \frac{\sin 2 u}{4} = \frac{\tan^{- 1} t}{2} + \frac{t}{2 (t^{2} + 1)} =

= \frac{\tan^{- 1} \sqrt{x^{2} - 1}}{2} + \frac{\sqrt{x^{2} - 1}}{2 x} + C

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