Quadrature for logarithmic weight: $ \int_0^1 f(x) x \log x\, dx.$

Question

Is there a standard way to evaluate (numerically) the integral

$$ \int_0^1 f(x) x \log(x) dx .$$

I was trying the substitution $u = -2\log(x)$, and then use Gauss-Laguerre quadrature. But it accumulates way too many points in $x=0,$ so it converges too slowly.

Any ideas/suggestions?

Answer 1

Collecting my comments into an answer, maybe helping future searches ...

As far as where to find a rule developed for working with a logarithmic singularity, a reference to a specific weighted Gaussian quadrature in DLMF:

• Have a look at Table 3.5.14.

In response to the OP's question on how these rules are calculated in practice:

• There’s a very good discussion in that DLMF chapter as well as in the monograph by Stroud and Secrest. But you might start simply with the description by Golub and Welsch. Best of luck!