THE METHOD OF U-SUBSTITUTION
The following problems involve the method of u-substitution. It is a method for
finding antiderivatives. We will assume knowledge of the following well-known,
basic indefinite integral formulas :
1.
2.
3.
4.
, where a is a constant
5.
6.
7.
, where k is a constant
8.
The method of u-substitution is a method for algebraically simplifying the form
of a function so that its antiderivative can be easily recognized. This method is
intimately related to the chain rule for differentiation. For example, since the
derivative of ex is
,
it follows easily that
.
However, it may not be obvious to some how to integrate
.
Note that the derivative of
is
can be computed using the chain rule and
.
Thus, it follows easily that
.
This is an illustration of the chain rule "backwards". Now the method of usubstitution will be illustrated on this same example. Begin with
,
and let
u = x2+2x+3 .
Then the derivative of u is
.
Now "pretend" that the differentiation notation
is an arithmetic fraction,
and multiply both sides of the previous equation by dx getting
or
du = (2x+2) dx .
Make substitutions into the original problem, removing all forms of x , resulting
in
=eu+C
= e x2+2x+3 + C .
Of course, it is the same answer that we got before, using the chain rule
"backwards". In essence, the method of u-substitution is a way to recognize
the antiderivative of a chain rule derivative. Here is another illustraion of usubstitution. Consider
.
Let
u = x3+3x .
Then (Go directly to the du part.)
du = (3x2+3) dx = 3(x2+1) dx ,
so that
(1/3) du = (x2+1) dx .
Make substitutions into the original problem, removing all forms of x , resulting
in
.
Most of the following problems are average. A few are challenging. Make
careful and precise use of the differential notation dx and du and be careful
when arithmetically and algebraically simplifying expressions.