Improper Integrals — Question 5

Question 5

Determine whether the integral converges: $$\int_2^\infty\frac{dx}{x\ln x}.$$

Question 5 – Solution

Step 1: Replace infinity with a limit. $$I=\lim_{b\to\infty}\int_2^b\frac{dx}{x\ln x}.$$ Step 2: Substitute $u=\ln x$. Then $$du=\frac{dx}{x}.$$ The bounds become $u=\ln2$ and $u=\ln b$, so $$\begin{align*} I&=\lim_{b\to\infty}\int_{\ln2}^{\ln b}\frac{du}{u}\\ &=\lim_{b\to\infty}[\ln u]_{\ln2}^{\ln b}\\ &=\lim_{b\to\infty}\left(\ln(\ln b)-\ln(\ln2)\right). \end{align*}$$ Step 3: Evaluate the limit. Since $\ln(\ln b)\to\infty$, the integral diverges. $$\boxed{\text{diverges}}$$