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 :
, where a is a constant
, where k is a constant
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 can be computed using the chain rule and is
. Thus, it follows easily that
. This is an illustration of the chain rule "backwards". Now the method of u-substitution 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
= e u + 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 u-substitution. 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
.