We need to split up the integral so we can deal with one badly-behaved limit of integration at a time. Let's split the integral at 0, since that's a nice easy number. Now we need to work out each of the simpler improper integrals (unless the first one diverges, in which case we'll be done!). Since *e*^{c2} approaches ∞ as *c* approaches -∞, the limit converges and The other improper integral is similar. Putting our results together, The original integral converges to 0. This makes sense, because in the graph it really looks like the total weighted area between the function and the *x*-axis should be 0. Yes, that was a lot of work to get 0 as an answer. Sometimes that's how it goes. And yes, it does make a lot of sense when you look at the graph that you would get zero for that integral. Alas, not all odd functions are that nice. The integral for example, doesn't exist. |