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Using series to solve definite integrals

  1. Dec 16, 2009 #1
    (this is not homework)

    Suppose I wanted to solve:

    [tex]\int log(x) log(x+1) dx[/tex] from 0 to 1.

    I would turn ln(x+1) into a series, namely, –∑(-1)^n * x^n / n

    Any ideas? Besides substituting, pulling out the n's, and using intgration by parts?
  2. jcsd
  3. Dec 16, 2009 #2
    what? why not just do it numerically? using the series representation of a fn then integrating terms by term doesn't help you in the least since if the integrated series had a simple representation then so would the antiderivative of the original function.

    for what it's worth here's the antiderivative:

    x - x (-1 + Log[x]) - Log[1 + x] + x (-1 + Log[x]) Log[1 + x] +
    Log[x] Log[1 + x] + PolyLog[2, -x]

    where PolyLog[2, -x] is the second order polylog function in -x.

    the definite integral turns out to be:

    2 - pi^2/12 - Log[4]
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