Question

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what is the answer of this question ​

Answer 1

Step-by-step explanation:

We have,

$\int \frac{1}{ {x}^{3} } \sqrt{ \frac{ {x}^{2} - 1}{ {x}^{2} } } dx$

$= \int \frac{1}{ {x}^{3} } \sqrt{1 - \frac{1}{ {x}^{2} } } dx$

$Let \: 1 - \frac{1}{ {x}^{2} } = t \\ \implies \frac{2}{ {x}^{3} } dx = dt$

$= \int \frac{ \sqrt{t }}{2} dt$

$= \frac{1}{2} \int \sqrt{t } .dt$

$= \frac{1}{2} . \frac{2 {t}^{ \frac{3}{2} } }{3} + c$

$= \frac{1}{3} {t}^{ \frac{3}{2} } + c$

$= \frac{1}{3} {(1 - \frac{1}{ {x}^{2} } )}^{ \frac{3}{2} } + c$

$= \frac{1}{3 {x}^{3} } {( {x}^{2} - 1 )}^{ \frac{3}{2} } + c$