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.