Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook