This integral is a classic example of how algebraic manipulation and appropriate substitutions can simplify complex expressions into standard integrable forms. The key strategy involves identifying patterns and using substitutions that transform the integrand into a more manageable structure, often involving terms like $x^2 + 1/x^2$ or $x - 1/x$.

1. Initial Setup and Analysis of the Integral

We are asked to evaluate the definite integral: $I = {{24} \over \pi }\int_0^{\sqrt 2 } {{{(2 - {x^2})dx} \over {(2 + {x^2})\sqrt {4 + {x^4}} }}}$

The integrand contains terms involving $x^2$ and $x^4$. The