Question

Determine whether the improper integral converges or diverges, and find the value of each that converges.

∫^-1_-[infinity] ln |x| dx

Answer 1

It diverges.

Explanation:

We are given the integral:
`\int\limits^(-1)_(-\infty) \ln |x| dx`

`\int\limits^(-1)_(-\infty) \ln |x| dx=\int\limits^(-1)_(-\infty) \ln (-x) dx=\\\\= \lim_(t \to \infty) \int\limits^(-1)_(-t) \ln (-x) dx= \lim_(t \to \infty)( x(\ln \left(-x\right)-1))|^(-1)_(-t)=1-\infty=-\infty`

So it is divergent.