Using series to solve definite integrals

[CDrappi]

(this is not homework)

Suppose I wanted to solve:

\int log(x) log(x+1) dx 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?

 

[ice109]

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]