Answer
-$ \frac{1}{3}.{ (1-\frac{1}{x^{2}})^{3/2}}$
Work Step by Step
$\int \frac{1}{x^{3}} \sqrt \frac{x^{2}-1}{x^{2}} dx $
Equation=$\int \frac{1}{x^{3}} \sqrt (1-\frac{1}{x^{2}}) dx $
Let u = $ (1-\frac{1}{x^{2}})$
du = $\frac{-2}{x^{3}}$
$-\frac{du}{2}$ = $\frac{1}{x^{3}}$
Putting values in equation
=$\int \sqrt u (-\frac{du}{2}) $
=-$\frac{1}{2} \int \sqrt u du$
=-$\frac{1}{2} \int \sqrt u du$
=-$\frac{1}{2} . \frac{u^{3/2}}{\frac{3}{2}}$
=-$\frac{1}{2} . \frac{2u^{3/2}}{3}$
=-$ \frac{u^{3/2}}{3}$
Putting values of u
=-$ \frac{1}{3}.{ (1-\frac{1}{x^{2}})^{3/2}}$
