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. ![](https://media1.shmoop.com/images/calculus/calc_indefint_improper_latek_131.png)
Now we need to work out each of the simpler improper integrals (unless the first one diverges, in which case we'll be done!). ![](https://media1.shmoop.com/images/calculus/calc_indefint_improper_latek_132.png)
Since ec2 approaches ∞ as c approaches -∞, the limit converges and ![](https://media1.shmoop.com/images/calculus/calc_indefint_improper_latek_133.png)
The other improper integral is similar. ![](https://media1.shmoop.com/images/calculus/calc_indefint_improper_latek_134.png)
Putting our results together, ![](https://media1.shmoop.com/images/calculus/calc_indefint_improper_latek_135.png)
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 ![](https://media1.shmoop.com/images/calculus/calc_indefint_improper_latek_136.png)
for example, doesn't exist. |