MHB How Does Integration to Infinity Work in Calculus?

[karush]

$$\Large{§8.8. 14} \\
\tiny\text {Leeward 206 Integration to Infinity}\\
\displaystyle
I=\int_{2 }^{\infty} \frac{1}{x\ln\left({x}\right)}\,dx \\
\begin{align}\displaystyle
u& = \ln\left({x}\right) &
du&=\frac{1}{x} \ d{x}
\end{align} \\
\displaystyle
I=\int_{2}^{\infty}\frac{1}{u} \,du = \ln\left({u}\right)\\
\text {back substittute u} \\
I= \ln\left({\ln\left({x}\right)}\right)\\
\text {don't see how this can go to }\infty \\
\tiny\text{ Surf the Nations math study group}$$

 

[MarkFL]

What do you get if you set it up as a limit?

 

[karush]

$$\Large{§8.8. 14} \\
\tiny\text {Leeward 206 Integration to Infinity}\\
\displaystyle
I=\int_{2 }^{\infty} \frac{1}{x\ln\left({x}\right)}\,dx \\
\begin{align}\displaystyle
u& = \ln\left({x}\right) &
du&=\frac{1}{x} \ d{x}
\end{align} \\
\displaystyle
I=\int_{2}^{\infty}\frac{1}{u} \,du = \ln\left({u}\right)\\
\text {back substittute u} \\
\displaystyle
I= \left[\ln\left(
{\ln\left({x}\right)}\right)
\right]_2^\infty = \left[2+\infty\right]=\infty\\
\tiny\text{ Surf the Nations math study group}$$