How to solve Integrals with natural logs

Question

How to solve the following integral with natural logs:

$$\int_0^e \frac 1{\ln (x^x)} dx$$

Can anyone help me to solve this.

Answer 1

This is an improper integral which diverges. $$\int_0^e \frac 1{\ln (x^x)} dx = \int_0^e \frac 1{x\ln (x)} dx $$

Upon substitution of $$ u=\ln x$$

We get $$ \int_{-\infty}^1\frac {1}{u} du =$$

$$\ln (|u| )|_{-\infty}^1 = -\infty$$