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## Homework Statement

The initial problem is to calculate

[tex]\int_{-\infty}^{\infty}\cos(x^{2})dx[/tex] using

[tex]t=x^{2}[/tex]

and then

[tex]t^{-\frac{1}{2}}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-tu^{2}}du[/tex]

## Homework Equations

## The Attempt at a Solution

I have, by transformation and use of the symmetry of both integrals come to

[tex]\int_{0}^{\infty}\frac{u^{2}}{1+u^{4}}du[/tex]

which is easily solveable using Mathematica.

Alas the solution provided by mathematica is absolutely non-obvious to me.

Any good idea for a substitution and/or other ways?

I've tried splitting the fraction into smaller parts, but got an integral with i in it, which is not allowed as this is real calculus.

Thanks in advance