Phong Do
Integration by Substitution
Successful integration of a problem is always a challenge for students taking Calculus. It is an understatement to
say that, integration is among the most difficult concepts in Calculus that the students will encounter. This section
is an attempt to provide a quick review of the integration by substitution – a method which is normally covered in
Calculus I. It would be helpful to review these before moving on further.
Basically, the method of substitution can be divided into 2 types:
o
The
u-substitution
o
The
miscellaneous substitution
At the very least, students who already had Calculus I
should be very familiar and comfortable in using the “u-
substitution”.
The best place to get to know the “
u-substitution
” starts with the Power Rule:
1
1
1
n
n
u
u du
C
n
n
+
+
+
≠ −
=
∫
Pay careful attention to the rule and you will note that, instead of
“x”
and
“dx”
,
we use
“u”
and
“
du”
. And if you
look more closely at ALL integration rules provided in your book and everywhere else, they all contain
“u”
and
“
du”.
So it is important that you know how to work with them. Below is a snapshot of several formulas from the
book that you should have been exposed to back in Calculus I.
To use the “
u-substitution
”,
you must make sure that the “u” and
the “du” are consistent.
If they are
NOT
consistent, see if
YOU
can
make them consistent. And to be successful, you must:
o
Recognize the pattern quickly
o
Determine the derivative quickly
The best way to learn the
u-substitution
is to study plenty of examples and then practice!
The easiest type of
u-substitution
problem is the ones involving polynomials. And that is where we will begin. As
you go through them, pay attention to how we get the
“u”
and
“du”
to be consistent.

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