integral calculate $f:[e,\infty)\to\mathbb{R}$

Question

$f:[e,\infty)\to\mathbb{R}$ $f(x)=\frac{1}{xlnx}$

I want to show $\int_{[e,\infty)} f d\lambda$ and I know it has to be $=\infty$ How can I do that?

Answer 1

By Monotone Convergence Theorem we know that \begin{align*} \int_{e}^{\infty}\dfrac{1}{x\ln x}d\lambda=\lim_{n\rightarrow\infty}\int_{e}^{n}\dfrac{1}{x\ln x}dx. \end{align*} The function $\dfrac{1}{x\ln x}$ is continuous on $[e,n]$, so the Lebesgue integral and Riemann integral coincide, we can use the Fundamental Theorem of Calculus to compute that \begin{align*} \int_{e}^{n}\dfrac{1}{x\ln x}d\lambda=\int_{e}^{n}\dfrac{1}{x\ln x}dx=\ln(\ln x)\bigg|_{x=e}^{n}=\ln(\ln n), \end{align*} now $n\ln n\rightarrow\infty$ as $n\rightarrow\infty$.