The exercise statement gives you the following:
. . . . .
So there is no "computation" or "figuring" to do. This is just something you are given. You are also given the integral expression for E(x) as being something that occurs in probability and statistics. For the origin of this expression, you'd likely need to study probability or statistics.
In other words, there's nothing to figure out or compute or develop or explain; you're just using what they've given you.
Note: There appears to be a typo in the solution, as the
in the original expression suddenly changes to
at the end of the worked solution to part (a), which of course would prevent the simplification to
. Draw in the square-root symbol, and the proof should be fine.