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I'm attempting a novel approach to some tough integration problems. I'm using the idea of series expansions to help integrate. In other words, I will attempt to approximate integration by integrating the series expansion of an integrand, rather than direct integration or standard numerical methods.

I believe I can approximate integration of a series very easily, compared to the other methods. However, there's a catch. I will use at least two different series expansions. One for the lower limit of integration, and one for the upper limit. Now, when I attempt to integrate these expansions, the constant of integration comes into play, and it's not obvious what it is. Since I am using at least two different series expansions, the constant of integration may differ for each expansion. So I'm wondering if there is an easy way to get the constants of integration without much more work. Any help, ideas, or suggestions are welcome.

EDIT

A few additional notes... I know ahead of time that the series will converge. I consider that I could integrate in sections, like quadrature, while still using the series to aid in integration. However, I am considering the idea of only using only the endpoints, with two different series. So the constants of integration would be different for each series. If I could somehow find them or find how they differ relative to one another, that would save me the trouble of breaking the integral into sections and using something akin to conventional numerical methods.

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    @André Nicolas: I'm interested in "matching". I'd like to hear basically any/all potential solutions to the problem. Please feel free to contribute - your help is much welcomed and appreciated!2011-08-08

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If you are talking about looking for the anti-derivative (though you also mentioned upper and lower limits), then you just write some constant $C$ after all your calculations. E.g. if $ f(x) = \sum\limits_{k=0}^\infty a_k x^k $ then $ F(x) = \int f(t)\,dt = \sum\limits_{k=0}^\infty\frac{a_k}{k+1}x^{k+1} +C. $ The idea of an anti-derivative is the following: $ \int\limits_{a}^bf(t)\,dt = F(b)-F(a) $ for any anti-derivative $F$ of the function $f$. Since any anti-derivative is determined up to a constant, you just should pick one anti-derivative before calculating $F(b)-F(a)$. In our example, you can put $C=1$ or $C=2$ and the result will stay the same.

Finally, to apply such a technique you should be careful since you cannot always integrate series in such a simple way - there are sufficient conditions to integrate series by parts.