Author:
Christopher Danielson

To demonstrate a special circumstance of integration using u-substitution.

A clever technique for u-substitution is introduced through a single example.

Tutorial

This packet assumes you are famiilar with the basic integration technique of *u*-substitution. It presents a demonstration of a clever tweak to that technique for dealing with an integral such as:

The standard u-substitution technique would be to let

But this doesn't resolve the x-squared part. What to do? Solve the *u* equation for *x*.

As in the following video...

In the video you are about to watch, you see me write (and hear me say)...

But this is not quite right. If x>0, it's right. But we haven't made that assumption. So let's make that assumption explicit.

**In the following video, let x≥0**.

Without this assumption, the correct statement is:

We regret the omission (and we thank Matt Dempsey for noticing it).

A clever u-substitution technique is demonstrated in this video.

Having made the pre-video disclaimer, the motivation for the technique actually came up in solving a problem that calls on students to find the area inside the loop shown in this packet's picture.

In that case, x is most definitely not greater than zero. Indeed, the whole integral will take place to the left of the y-axis, with negative x values. In that case, we'll use:

And then we'll double the area under the curve between x=-3 and x=0 to get the area of the loop.

If you don't use -x, you'll get the opposite of the solution you seek.