Prove convergence of this generalized integral

Question

Prove the convergence of $$\int_0^1 \left[\ln\left(1+\frac1x\right)\right]^a\mathrm dx$$ for $ a>0$.

Answer 1

Setting $1/x=y$, this is equal, up to a costant, to $$\int_1^\infty\frac{\left(\ln(1+y)\right)^a}{y^2}\, \mathrm{d}y<<\int_1^\infty \frac{\sqrt{y}}{y^2}\mathrm{d}y,$$ which converges for all $a>0$.