MATH 164, Spring 2001 Due Date: Name(s):
Project 8.6: Techniques of Integration
Objective
To illustrate the strengths and weaknesses of Maple in evaluating integrals.
Narrative
If you have not already done so, read Section 8.6 in the text.
This project involves two tasks. The first illustrates how easily Maple integrates some expressions (such
as partial fractions), and how Maple can be used to check some of the complicated intermediate work you
might have to do by hand when doing your homework. The second illustrates that while Maple's ability to
evaluate integrals is powerful, it is not so great that we can dispense with learning techniques of integration.
Tasks
1. Type the command lines in the left-hand column below into Maple in the order in which they are listed.
These commands are aimed at finding
dx
x3 - x
.
> # Project 8.6: Techniques of Integration
> # Task 1
> restart; Clear Maple's memory.
> int(1/(x^3-x),x); Find
dx
x3 - x
.
Maple handles this integral pretty fast. Let's see how Maple can be used to check some of the complicated
intermediate work you might have to do by hand when doing your homework. Continue by typing the
command lines in the left-hand column below into Maple in the order in which they are listed.
> convert(1/(x^3-x),parfrac,x); Find the partial fraction decomposition of the integrand
1/(x3
- x).
> int(%,x); Integrate the result.
> diff(%,x); Check our answer by differentiating it.
> simplify(%); Simplify to make comparison with the original integrand
easier.
2. a) Type the command lines in the left-hand column below into Maple in the order in which they are
listed. These commands are aimed at finding
arcsin x
(1 - x2)3/2
dx.
> # Task 2
> restart; Clear Maple's memory.
> MyInt := int(arcsin(x)/((1-x^2)^(3/2)),x); Let MyInt =
arcsin x
(1 - x2)3/2
dx.
Note that Maple can't evaluate
arcsin x
(1 - x2)3/2
dx: it would have if it could have in response to the third
command line you typed!
At this time, make a hard-copy of your typed input and Maple's responses. Then, ...
b) use the substitution x = sin u to evaluate
arcsin x
(1 - x2)3/2
dx by hand.
Comments
The moral of Task 1 is that Maple is a very powerful tool when it comes to techniques of integration. The
moral of Task 2 is that as powerful as Maple is, there are (and always will be, some fairly simple) integrals
it cannot evaluate, and hence we cannot rely on it (to the extent that we can avoid learning techniques of
integration).