What is the integral from 1/e to e of 1/(xln(x)) ?

The integral from 1/e to e of 1/(xln(x)) is diverges

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integral from 1/e to e of 1/(xln(x))

\int_{\frac{1}{e}}^{e} \frac{1}{x ln ( x )} d x

diverges

Solution steps

\int_{\frac{1}{e}}^{e} \frac{1}{x ln ( x )} d x

Find undefined (singularity) points:

x = 1

If exist

b, a < b < c, f (b) =

undefined

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{\frac{1}{e}}^{1} \frac{1}{x ln ( x )} d x + \int_{1}^{e} \frac{1}{x ln ( x )} d x

\int_{\frac{1}{e}}^{1} \frac{1}{x ln ( x )} d x =

diverges

If part of the expression diverges, then entire expression diverges

= diverges

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What is the integral from 1/e to e of 1/(xln(x)) ?
The integral from 1/e to e of 1/(xln(x)) is diverges