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# 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.

Problem   Find

Now is just .

This looks ripe for a u-substitution. Let's take , so and

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

Next: Glossary: A Quick Guide to the Mathematical Jargon: Up: Doing that Calc Thing Previous: Working with other Bases
Joel Hass
1999-05-26