CHAPTER 5. INTEGRALS REVIEWED
5.5
5
U -substitution
Here is a brief outline of the technique
of U -substitution.
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0. You are given an integral h(x) dx where h(x) is some complicated function
of x.
1. Fill in the following
u=
du = (
something
derivative of u
(you get to pick this)
) · dx
(you don’t get to pick this)
Most often you pick u to equal something inside of another function
2. Fill in the following
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translate or cancel x’s and dx
h(x) dx = . . . . . . . . . . . . . . . = f (u) du.
.
Make sure that all the x’s (including dx) cancel by the last step; whatever
you’re left with, call it f (u).
3. Find the anti-derivative
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f (u) du = F (u)
4. Inside of F (u), replace u with the same “something” involving x, that you
picked in step 1.
Comments. When you are practicing u-substitution here are some good bits of
advice and rules of thumb to keep in mind.
To set u = something you should be willing to take a guess, try setting u equal
to something, and then take the derivative, and see if you can get rid of all the x’s
using u and du.
Above I wrote “translate or cancel all the x’s and dx”. Different people have
different approaches to this step, but they all produce the same result; approach it
whichever way you want. Here’s a brief description of two approaches:
1. just circle those x’s that are part of u and those that are part of du, and
replace those parts with u and du,
2. solve for dx, and replace it with a formula involving du and x’s, and then
cancel any remaining x’s.
Whichever approach you take, you should get
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h(x) dx = f (u) du
and should be able to double check your work as follows. If you start with the
right hand side, and substitute x’s back in for u and du, you should get the original
integral, the one on the left.
When using u-substitution, it helps to know what sort of targets you might have
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for
F (u) du. In other words, you want to have a list of integrals that are known,
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and then see if you can turn
h(x) dx into one of known integrals.