A clever integration technique

A clever integration technique


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

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

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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).

Demonstrating the technique

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

Now, that said...

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.