Integrals and the Natural Log

We know, you're expecting us to begin this section by telling you what equals. But that's not really what's important here. If you really must know, it equals , but few professors expect you to remember that (although it can be worked out using integration by parts, a section coming up shortly, and there, it makes a good problem.)

No, in this section we would just like to reverse the formula

to obtain one of the most famous formulas in all calculus:

There are a couple of things to notice here. First, by integrating a function
1/*x* that appears to have nothing whatsoever to do with *e* , wouldn't
recognize *e* if *e* bit it on the nose, we get the log base *e*. Just another
way of saying,``*e*, you're mighty special."

Second, you will notice that somehow, the *x* in the solution picked up absolute
value signs. That's because the natural log function is only defined for
positive values of *x*, so if *x* were negative and we didn't have the absolute
value signs, it would make no sense. Should you worry about the absolute value
signs? If you're going for the A+, YES, otherwise forget about them.

This looks ripe for a

Now, was that as good for you as it was for us?