Big O, Little O and Logarithm Integral

Hello everyone,

I've spent the last hours trying to understand Big O and little O notations in depth. Usually the limit of the quotient method works but here, I'm dealing with integrals which seems to make things harder.

integral(dt/ln(t),t=2..sqrt(x))=O(sqrt(x)) when x->+∞

integral(dt/(ln(t))^2,t=sqrt(x)..x)=o(integral(dt/ln(t),t=sqrt(x)..x) when x->+∞

I'd really appreciate some tips or the general method on proving these two assertions. Sorry for not knowing Latex, I'll get to it soon ^^

Thanks for any help!