Notice that the integral goes from 0 to infinity. Since the exponential distribution measures the time until first occurrence of an event, it cannot be negative.

To solve we must integrate by parts. Let u=x and dv equal the rest of the integral. Then, we get:

The tricky part here is to recognize that e^-x goes to zero rapidly as x goes to infinity. Exponential decay is so rapid that even x e^-x goes to zero as x goes to infinity (despite the fact that x in this equation is going to infinity).