Find $\lim_{x \rightarrow 1} \int_{x}^{ x^{2}} \frac{1}{\ln(t)}\,\mathrm dt$

Question

Find: $$ \lim_{x \rightarrow 1} \int_{x}^{ x^{2}} \frac{1}{\ln(t)}\,\mathrm dt $$

My calculations lead to 1, but as far as I know, this is not correct result.

Answer 1

By setting $x=e^u$, then $t=e^s$, we have to compute: $$ \lim_{u\to 0}\int_{e^u}^{e^{2u}}\frac{dt}{\log t}=\lim_{u\to 0}\int_{u}^{2u}\frac{e^{s}}{s}\,ds =\lim_{u\to 0}\int_{u}^{2u}\frac{1+s+o(s)}{s}\,ds=\color{red}{\log 2}.$$