Using series to solve definite integrals

In summary, the conversation discusses how to solve the integral of log(x) log(x+1) from 0 to 1. The suggestion is to use the series representation of ln(x+1), but this does not provide a simple solution. The antiderivative and definite integral are also mentioned.
  • #1
CDrappi
15
0
(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?
 
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  • #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]
 

Related to Using series to solve definite integrals

What is the concept of using series to solve definite integrals?

The concept involves using infinite series, such as Taylor or Maclaurin series, to approximate the value of a definite integral. This is useful in situations where the integral cannot be evaluated directly.

How is a Taylor series used to solve a definite integral?

A Taylor series is a representation of a function as an infinite sum of terms. By integrating both sides of the series, we can obtain a new series that can be used to approximate the original integral.

What are some common techniques for using series to solve definite integrals?

Some common techniques include using algebraic manipulation, substitution, and integration by parts to transform the integral into a form that can be evaluated using a known series.

Are there any limitations to using series to solve definite integrals?

Yes, there are limitations. Series can only be used to approximate the value of definite integrals, and the accuracy of the approximation depends on the number of terms used in the series. Additionally, series may not be applicable to all integrals and may not converge for certain functions.

How does using series to solve definite integrals compare to other methods?

Using series to solve definite integrals can be a powerful tool, but it is not always the most efficient or accurate method. Other techniques, such as numerical integration or the fundamental theorem of calculus, may be more appropriate for certain integrals.

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