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Evaluating an Integral using Jordan's Lemma

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The problem is:

Evaluate the integral from 0 to INF of:
[(x^(1/3))*(ln x)]/(x^2 +9) dx

by using f(z)= [(z^(1/3))*(Log z)]/(z^2 +9), with
-pi/2 < Log z < 3pi/2.
Also, with z^(1/3)= e^[(1/3)Log z].

We are to use the curve C:
from -R to -p, -p to p around origin, p to R, and Cr from 0 to pi.

Many thanks in advance for your help. This is TOUGH one!

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An integral is evaluated using Jordan's lemma. A diagrm is included in this well presented solution.

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