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It is well known that many integrals have no closed form solutions, normally what you would do is solve them numerically.

My question is if there are algorithms that give you good closed form approximations instead.

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If you can replace the integrand you're interested in with some function that is easy to integrate, and looks very much like the original integrand, then certainly... – J. M. May 4 '12 at 12:51
up vote 1 down vote accepted

You can write

$$f(x) = a_0x^{b_0} + a_1x^{b_1} + ... + o(x^{b_n})$$


$$\int f(x) dx = a_0\frac{x^{b_0+1}}{b_0+1} + a_1\frac{x^{b_1+1}}{b_1+1} + ... + o(x^{b_n+1})$$

(however be carefull with $b_k=-1$)

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