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I'm intested in learning how to get a sum in difference calculus. I'm willing to settle, at least for now, in learning how to integrate the following integral in calculus:

$\int(a^x(1-a^x)(1-a^x))dx$

The sum that I'm really interested in learning how to get, hopefully using difference calculus, is: $\sum_x{a^x(1-a^x)(1-a^x)}$

Even better still, extend this technique to include more $(1-a^x)$, and add limits of integration or summation.

_EDIT_

I'd like to do so without multiplying everything out, if possible

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If you have more factors $(1-a^x)$, you can still multiply out using the binomial theorem, and integrate termwise to get an answer in the form of a sum involving binomial coefficients. I doubt that these answers can be simplified very much. For example, trying Integrate[E^x (1-E^x)^n, x] // Factor in Mathematica with different values of $n$ doesn't show any obvious factorization pattern. –  Hans Lundmark Mar 15 '11 at 18:10

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up vote 2 down vote accepted

Well, integration is easy: just multiply everything out to get a sum where each term has the form $a^{kx} = e^{(k \ln a)x}$, then integrate term by term.

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Summation is just as easy; sums of the form $a^{kx}$ are geometric series and converge absolutely for $|a|<1$. –  mjqxxxx Mar 15 '11 at 16:01

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