[Math] Determine whether the integral is convergent or divergent. $\int^{\infty}_1 81\frac{\ln(x)}{x}dx$

Question

Determine whether the integral is convergent or divergent.$\int^{\infty}_1 81\frac{\ln(x)}{x}dx$

My answer is $\infty-0=\infty$

But I am unsure whether it is convergent or divergent.

Answer 1

The constant $81$ is irrelevant, so we can just consider $$ \int_{1}^{\infty}\frac{\ln x}{x}\,dx $$ The integral converges if and only if $$ \int_{e}^{\infty}\frac{\ln x}{x}\,dx $$ converges. Since $\ln x\ge1$ for $x\ge e$, we have $$ \frac{\ln x}{x}\ge\frac{1}{x} $$ However $$ \int_e^\infty\frac{1}{x}\,dx $$ does not converge, because $\lim_{x\to \infty}\ln x=\infty$, therefore also the given integral is divergent.