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  4. ∫ (x4-4/x2 √4+x2+x4) d x equals-

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Q. $\int \frac{x^4-4}{x^2 \sqrt{4+x^2+x^4}} d x$ equals-

Integrals

Solution:

$\int \frac{x^4-4}{x^3 \sqrt{\frac{4}{x^2}+1+x^2}} d x=\int \frac{x-4 x^{-3}}{\sqrt{4 x^{-2}+1+x^2}} d x$
$=\frac{1}{2} \int t^{-1 / 2} d t $ [put $\left.t=4 x^{-2}+1+x^2\right]$
$=\sqrt{\frac{4}{x^2}+1+x^2}+C=\frac{\sqrt{4+x^2+x^4}}{x}+C$