Estimate of a function on different ranges

Question

I am currently reading Montgomery's paper on Pair Correlation Conjecture. In his paper, he defines some function $R(x, t)$ and goes on to show that \begin{equation} \int_{0}^{T} |R(x, t)|^2 \ dt = \begin{cases} (1 + o(1))(T/x^2) \log^2 T & \text{if } 1 \leq x \leq (\log T)^{3/4}, \\ o(T \log T) & \text{if } (\log T)^{3/4} < x \leq (\log T)^{3/2} \\ (1 + o(1))T \log x & \text{if } (\log T)^{3/2} < x \leq T/\log T. \end{cases} \end{equation} How can we conclude from this that $$ \int_{0}^{T} |R(x, t)|^2 \, dt = (1 + o(1))(T/x^2) \log^2 T + o(T \log T) + (1 + o(1))T \log x. $$ I thought this is because in each of the three ranges, the terms for other ranges get engulfed in the error term for that particular range but this is not the case. For instance, the term $o(T \log T)$ does not gets engulfed in the error term $o((T/x^2) \log^2T)$ in the range $1 \leq x \leq (\log T)^{3/4}$ as $$ \frac{o(T \log T)}{(T/x^2) \log^2 T} \leq \frac{x^2 o(T \log T)}{T \log^2 T} \leq o((\log T)^{1/2}) $$
when in fact we wanted it to be $o(1)$. Any help or suggestions will be highly appreciated.