Find an asymptotic formula for $\sum\limits_{n\leq x} d(n)\log n$ Please help me to find the asymptotics for the sum described above: 
$$\sum\limits_{n\leq x} d(n)\log n.$$
This is a problem in analytic number theory.

By the Dirichlet's hyperbola method we have:
$$ \sum_{n\leq x}d(n) = x\log x+(2\gamma-1)x+O(\sqrt{x})$$
hence Abel's summation formula gives:
$$\sum_{n\leq x}d(n)\log n = x\log^2 x+(2\gamma-1)x\log x-\int_{1}^{x}\left(\log u+(2\gamma-1)\right)\,du+O(\sqrt{x}\log x) $$
that simplifies to:
$$\sum_{n\leq x}d(n)\log n = x\log^2 x+(2\gamma-2)x\log x+O(x).$$
