If $f$ is Lebesgue integrable, is $f^2$ also Lebesgue integrable?

Question

Is there an example where $f$ is Lebesgue-integrable but $f^2$ isn't?

Answer 1

Try for example $$ \matrix{(0,1] & \longrightarrow & {\bf R} \cr x & \mapsto & {1\over \sqrt{x}} \cr} $$