Find the exact volume of the solid created by rotating the region bounded by $f(x)=\frac1{\sqrt x\ln x}$ and the $x$-axis on the interval $[2, ∞)$. State the method of integral used.
My issue specifically is that I have no clue how to integrate the integral because I am using the disk method and the integral ends up being $$\int_2^\infty\pi\cdot\frac1{x(\ln(x))^2}\,dx$$ How would I go about solving this because I am unsure how to integrate that integral.
I see $x(\ln x)^2$ in the denominator, which suggests that the antiderivative involves $\frac1{\ln x}$. And indeed: $$\int_2^\infty\pi\cdot\frac1{x(\ln(x))^2}\,dx=\pi\left[-\frac1{\ln x}\right]_2^\infty$$ Since $\lim_{x\to\infty}\frac1{\ln x}=0$, we get $$\pi\left[-\frac1{\ln x}\right]_2^\infty=\pi\left(0-\left(-\frac1{\ln2}\right)\right)=\frac\pi{\ln2}$$