Average lifetime

You might think that the half-life would also be the average lifetime of an atom, since half the atoms' lives are shorter and half longer. But the half whose lives are longer include some that survive for many half-lives, and these rare long-lived atoms skew the average. We can calculate the average lifetime as follows:

Using the convenient base-e form again, we have

This integral is of a form that can either be attacked with integration by parts or by looking it up in a table. The result is

and the first term can be ignored for our purposes because it equals zero at both limits of integration. We end up with

which is, as expected, longer than one half-life.