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$\large \int \frac{x dx}{\sqrt{x^{4}-1}}$

lets take $\large z=x^{2}$

              $\large \frac{\mathrm{d}z }{\mathrm{d} x}=2x$

               $\large \frac{dz}{2}=xdx$

substituting it in the integral,

$\large \int \frac{dz}{2\sqrt{z^{2}-1}}$

=$\large \frac{1}{2}\ln\left | z+\sqrt{z^{2}-1} \right |+c$          as,  $\large \left [ \int \frac{dx}{\sqrt{x^{2}-a^{2}}} \right ]=\ln \left | x+\sqrt{x^{2}-a^{2}} \right |+c$

=$\large \frac{1}{2}\ln\left | x^{2}+\sqrt{x^{4}-1} \right |+c$

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