An improper integral — from The Math Connection group in LinkedIn

A teacher posted a question on the Math Connection group in LinkedIn (here), how to compute the integral ∫-11dx/[1-exp(x1/3)].

The question was answered in one of the comments and I really like that solution. Here it is:

First, note that the integral exists, since the integrand

f(x) is O(x-1/3) near 0.

Now, since 1/(1-exp(-t)) can be written


we have f(-x) = 1 – f(x).

Hence, the integral from -1 to 0 can be transformed into an integral from 0 to 1 which, when added to the positive half of the original integral, will be just the integral of unity, and the final value is 1.

Published in: on 13/09/2013 at 09:49  Leave a Comment  

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