 In this video, we'll work through two examples of improper integrals that involve an infinite limit of integration. Consider the curve f of x equals x e to the negative x squared over the interval 0 to infinity, as shown here. Integrating this function over that interval, we write the integral from 0 to infinity of x e to the negative x squared dx using limit notation, limit as b approaches infinity from 0 to b, x e to the negative x squared dx. I'm going to pop out of this definite integral to integrate x e to the negative x squared using u substitution. If I let u be negative x squared, du dx is negative 2x, so x is equal to negative one-half du dx. And I have negative one-half e to the u du, which then gives me negative one-half e to the u plus c, which is negative one-half e to the negative x squared plus c. So I'm going to come back to this definite integral. This is limit as b approaches infinity of negative one-half e to the negative x squared from 0 to b using the fundamental theorem of calculus. We evaluate limit as b approaches infinity of negative one-half e to the negative b squared plus one-half e to the zero. Now as b approaches infinity, this term right here goes to zero and this ultimately equals one-half. So since this limit exists, this integral converges and in fact it converges to one-half. Consider a second example, the integral from negative infinity to zero of e to the negative 2x dx. We rewrite this integral using limit notation as the limit as a approaches negative infinity, the integral from a to zero of e to the negative 2x dx. Evaluating the integral, finding the antiderivative, we find this is the limit as a approaches negative infinity of negative one-half e to the negative 2x evaluated from a to zero. This is the limit as a approaches negative infinity of negative one-half e to the zero plus one-half e to the negative 2a. Now since a is approaching negative infinity and a appears in this negative exponent, we see that this limit equals infinity or approaches infinity since the limit does not exist, this integral diverges.