$f:[0,\infty)\longrightarrow[0,\infty)$ continuous, non negative.
prove or give a counter example
1) if ${\displaystyle \intop_{1}^{\infty}f\left(x\right)dx}$ exists then f is bounded.
2) if ${\displaystyle \intop_{1}^{\infty}f\left(x\right)dx}$ exists then ${\displaystyle \intop_{1}^{\infty}f^{2}\left(x\right)dx}$ exists as well.
I think both claims are wrong, but all the counter examples I could think of didn't work out.