It looks like this integral probably converges, since f (x) looks a lot like e^{-x}, and we know

converges.

We have to be careful though. It's true that

e^{-x} – 5 < e^{-x},

but e^{-x} – 5 isn't bounded below by 0, so the integral could diverge to -∞.

In fact that's exactly what happens. We have to make sure that our function is also bounded below by something that converges. Otherwise the integral could be equal to -∞ as was the case here.