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How would you deal with such an integral? Any tips will be appreciated!

$\int_0^{\infty} \left[\prod_{k=1}^K \sum_{j=0}^k a_j x^j\right] f(x) \, dx$

$a_j$ is a constant not depending on $x$, $f(x)$ is some function of $x$. My question is how to deal with the integral of the product...

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    @did oops, good point.2012-10-21

1 Answers 1

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Obviously first multiplier is a polynomial of some degree $N=\sum_{k=0}^K k=K(K+1)/2$. Call this polynomial $\sum_{i=0}^{N} b_i x^{i}$. Then you get $ \int_0^{\infty} \left[\prod_{k=1}^K \sum_{j=0}^k a_j x^j\right] f(x)dx= \int_0^{\infty} \left[\sum\limits_{i=0}^{N} b_i x^{i}\right] f(x) dx= \sum\limits_{i=0}^{N} b_i\int_0^{\infty} x^{i} f(x) dx $ Hence to attack this integral it is enough to

  • Compute integrals $\int_0^{\infty} x^{i} f(x) dx$ for all $i=0,\ldots,N$
  • Compute coefficients $b_i$, this is very messy.
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    @user7064 even if you get explicit formula for $b_i$ it will be also very messy. So horrible mess is unavoidable in this approach.2012-10-21