I want to evaluate the improper integral $\int\limits_{0}^{\infty}\frac{x^{1/4}}{1+x^3}\, dx$ via residue theorem but something odd is happening.
When I use the key-hole contour where I integrate above/below the postive real axis, I end up getting that the real and imaginary part of the integral is $-\int\limits_{0}^{\infty}\frac{t^{1/4}}{1+t^3}dt + \int\limits_{0}^{\infty}\frac{t^{1/4}}{1+t^3}dt*i $
When I compute the contour via residues I get answers that not only do not match up to numerical calculation but I but have different real and imaginary scaler values.
The 3 roots of $1+z^3$ are $-1, 1/2+\frac{\sqrt{3}}{2i}, 1/2-\frac{\sqrt{3}}{2}*i $
And residue values computed at each are:
for $-1$, $\frac{\sqrt{2}}{6}(1+i)$
for $1/2 + \sqrt(3)/2i$, $\frac{-(\sqrt{3}-1)\sqrt{2}}{12} - \frac{(\sqrt{3}+1)\sqrt{2}}{12}i$
for $1/2 - \sqrt(3)/2i$, $\frac{-(\sqrt{3}-1)\sqrt{2}}{12} + \frac{(\sqrt{3}+1)\sqrt{2}}{12}i$
Now clearly the sum of these multiplied by $2\pi*i$ will not have real and imaginary parts which are scaler multiples of each other.
What did I do wrong?