I read in a book that the condition $\int |f(x)|^2 dx <\infty$ is less restrictive than $\int |f(x)| dx <\infty$. That means whenever $\int |f(x)| dx$ is finite, $\int |f(x)|^2 dx$ is also finite, right?
My understanding is that $|f(x)|$ may have a thick tail to make the integral blow up, but $|f(x)|^2$ may decay quickly enough to have a finite integral. Can someone give me an example that $\int |f(x)| dx=\infty$ but $\int |f(x)|^2 dx <\infty$. Suppose $f(x)$ is an absolutely continous function and bounded on $(-\infty, \infty)$.