I am studying the prime number theorem and related stuff and was trying to solve this following problem:
Suppose there exists a constant $c$ such that $\psi(x) = x + (c + o(1))\frac{x}{\log x}$ as $x \rightarrow \infty$.
Deduce that $\int_2^x \frac{\psi(t)}{t^2} dt = \log x + (c + o(1))\log\log x$ as $x\rightarrow \infty$.
I can understand that I have to split the interval of integral from $2$ to $\sqrt x$ and $\sqrt x$ to $x$, and probably use the given relation in the second integral, but I am kind of getting all stuck. And also, a bit lost on the first integral. Any hints or explanations/suggestions please? Thanks