- #1

FeDeX_LaTeX

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## Homework Statement

Using contour integration, evaluate ##\int_{0}^{\infty} \frac{\sqrt{x}}{x^3 + 1} dx##.

## The Attempt at a Solution

Normally what I try to do in these problems is consider the upper half of a semi-circle from -R to R in the complex plane, as R goes to infinity. In doing so, I've found the residues as:

##\frac{1}{3}e^{i\frac{\pi}{2}}## at the pole ##z = e^{-i\frac{\pi}{3}}##;

##\frac{1}{3}e^{-3i\frac{\pi}{2}}## at the pole ##z = -1##;

##\frac{1}{3}e^{-i\frac{\pi}{2}}## at the pole ##z = e^{i\frac{\pi}{3}}##.

Summing and multiplying by ##2 \pi i## yields ##-\frac{2 \pi}{3}##.

However, what I've noticed is that my (real) integral is only defined for x ≥ 0. Does this mean I should consider instead the upper half of a semi-circle from 0 to R as R goes to infinity, so that only the pole ##z = e^{i\frac{\pi}{3}}## lies in my contour? The problem is, when I do that, I end up getting ##\frac{2 \pi}{3}##... still not the correct answer which is apparently ##\frac{\pi}{3}##.

What am I doing wrong here?