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A clever integration technique

A clever integration technique

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.

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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:

integral square root of x squared open parentheses x plus 3 close parentheses end root d x

The standard u-substitution technique would be to let

u equals x plus 3

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)...

square root of x squared end root equals x

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:

square root of x squared end root equals open vertical bar x close vertical bar

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:

square root of x squared end root equals negative x

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.