,
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 = x^{2}+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 = x^{3}+3x .
Then (Go directly to the du part.)
du = (3x^{2}+3) dx = 3(x^{2}+1) dx ,
so that
(1/3) du = (x^{2}+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.
Click HERE to see a detailed solution to problem 1.
Click HERE to see a detailed solution to problem 2.
Click HERE to see a detailed solution to problem 3.
Click HERE to see a detailed solution to problem 4.
Click HERE to see a detailed solution to problem 5.
Click HERE to see a detailed solution to problem 6.
Click HERE to see a detailed solution to problem 7.
Click HERE to see a detailed solution to problem 8.
Click HERE to see a detailed solution to problem 9.
Click HERE to see a detailed solution to problem 10.
Click HERE to see a detailed solution to problem 11.
Click HERE to see a detailed solution to problem 12.
Click HERE to see a detailed solution to problem 13.
Click HERE to see a detailed solution to problem 14.
Click HERE to see a detailed solution to problem 15.
Click HERE to see a detailed solution to problem 16.
Click HERE to see a detailed solution to problem 17.
Click HERE to see a detailed solution to problem 18.
Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :