Show that is convergent. Solution We can't evaluate the integral directly because the antiderivative of e"* is not an elementary function. We write ex dx = ex dx +/" ( dx and observe that the first integral on the right-hand side is just an ordinary definite integral. In the second integral we use the fact that for x 2 1 we have x 2 x, so -x s -x and therefore e"* sex. (See the figure below.) y = exs y= ex X The integral of ex is easy to evaluate as follows. ex dx = lim edx = lim (e-] - e-) = t -+ 00 Thus, taking f(x) = e-* and g(x) = e-* in the comparison theorem, we see that e - * dx is convergent. It follows that e- * dx is convergent