Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $ Evaluate the integral, 
$$ \int_{0}^{1} \ln(x)\ln(1-x)\,dx$$ 
I solved this problem, by writing power series and then calculating the series and found the answer to be $ 2 -\zeta(2) $, but I don't think that it is best solution to this problem.   I want to know if it can be solved by any other nice/elegant method. 

You could expand
$\ln(1-x)
=-\sum_{n=1}^{\infty} \frac{x^n}{n}
$
and evaluate
$\int_0^1  x^n \ln x\,dx$,
probably by an induction
via integration by parts.
From your description,
you may have already done this.
It sure is easier to 
write this than to do it.
