Can we apply the Fundamental theorem of integration on $f(x)=\frac{1}{x}$

Question

So, we're given $f(x)=\frac{1}{x}$ in $[-1;1]$.

AFAIK, the theorem only works if $f(x)$ is continuous, which in this case it isn't for $x=0$. So we cannot use the theorem.

If we just integrate it:

$\int_{-1}^1 \frac{1}{x}\,dx=ln|1|-ln|-1|=0$

Is this right? I'm just getting started and need a little help!

Answer 1

It is not right in because you integrated over an interval with zero as an x value. This has 1/0 which is undefined. If you are in a higher level course you could use the Cauchy principal value and get 0 but, if you are just starting it is wrong. This is one of those questions that depends on the course.