Evaluate $\int_1^\infty \frac {43\ln(x)}{x} dx $ if it exists

Question

Question: Evaluate if exists $$\int_1^\infty \frac {43\ln(x)}{x} dx $$

My answer: By using integration by substitution, $$\int_1^\infty \frac {43\ln(x)}{x} dx =[43(\ln(x))^2]_1^\infty=\infty-0=\infty$$ So this integral does not exist. Does this seem correct? Any help is much appreciated.

Answer 1

Yes, your result is correct (modulo some constant issues in the antiderivative and the bounds). Note that without computing, we can see that for $x>e$, $\ln x>1$. So $$ \int_{e}^\infty \frac{\ln x}{x}\mathrm dx\geq \int_{e}^\infty \frac{1}{x}\mathrm dx $$