Indefinite integral

Question

I want to integrate $\frac{1}{(1+x^4)}$ from zero to infinity, set $z_0,z_1,z_2$ and $z_3$ to be the roots of: $1+x^4$ Using Cauchy integral formula, on which path I should integrate?

Answer 1

Symmetry is your friend. This integrand is symmetric under the rotation $z \to i z$, so you might try going from $0$ to $+\infty$ along the real axis and coming back in along the positive imaginary axis. To make it a finite closed contour, you'll want to make that into a quarter-circle: from $0$ to some large $R$, then along a circular arc to $iR$, and back to $0$.

Answer 2

Your function is an even function... I would go from -infinity to infinity using the standard contours for that, and divide the result in two.