MATH 164, Spring 2001 Due Date: Name(s):
Extra Project 8.4a: Partial Fractions
Objective
To illustrate how Maple can be used to integrate a rational function by partial fractions.
Narrative
One of the challenging aspects of integration by partial fractions is the algebra involved in obtaining the
partial fraction decomposition of the integrand. In this project we illustrate how Maple can be used to find
x4
- 2x2
+ 4x + 1
x3 - x2 - x + 1
dx.
Our reason for doing this is two-fold: First, it might help you with your MATH 164 homework. Second, it
is important to learn how to do a partial fraction decomposition of a rational function since doing so is a
basic skill required in solving differential equations by using Laplace transforms.
Task
Type the command lines in the left-hand column below into Maple in the order in which they are listed.
The effect of each command is described in the right-hand column for your reference. Your lab report will
be a hard copy of your typed input and Maple's responses.
> # Project 8.4a: Partial Fractions
> restart;
First let num and den denote the numerator
and denominator of the fraction.
> num := x^4-2*x^2+4*x+1; den := x^3-x^2-x+1;
Next find the quotient and remainder.
> quo(num,den,x); rem(num,den,x);
Now factor the denominator.
> factor(den);
So the fraction we need to break into
parts is ...
> 4*x/((x+1)*(x-1)^2) = A/(x-1) + B/(x-1)^2 + C/(x+1);
To do this first clear denominators ...
> (x+1)*(x-1)^2*%;
> eq := simplify(%);
Now equate the constant on the lefthand
side of eq with that on the righthand
side ...
> eq0 := coeff(lhs(eq),x,0) = coeff(rhs(eq),x,0);
the coefficient of x on the left-hand side
of eq with that on the right-hand side ...
> eq1 := coeff(lhs(eq),x,1) = coeff(rhs(eq),x,1);
and the coefficient of x2
on the left-hand
side of eq with that on the right-hand
side ...
> eq2 := coeff(lhs(eq),x,2) = coeff(rhs(eq),x,2);
We then solve eq0, eq1, and eq2 for A,
B, and C ...
> solve({eq0,eq1,eq2},{A,B,C});
Let's check our work ...
> simplify(1/(x-1)+2/(x-1)^2-1/(x+1));
And finally let's compute the integral ...
> int(x+1+1/(x-1)+2/(x-1)^2-1/(x+1),x);
Don't forget that we can always check
our answer by differentiating.
> simplify(diff(%,x));
Comments
The purpose of this project was to illustrate how Maple can be used to help you check he computations
that you would otherwise do by hand in completing your homework. If you just want to check whether you
answer is correct, you can either: a) see what you get when you ask Maple to add the partial fractions, or
b) use the Maple command parfrac() which computes the partial fraction decomposition of the argument
(see Maple's Help for further details).