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18.01 Single Variable Calculus
Fall 2006
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HEAVISIDE'S
COVERUP
METHOD
The eponymous method was introduced
by
Oliver Heaviside as a fast way to do a decom
position into partial fractions. In 18.01 we need the partial fractions decomposition in order
to integrate'rational functions (i.e., quotients of polynomials). In 18.03, it will be needed as
an essential step in using the Laplace transform to solve differential equations, and in fact
this ývas more or less Heaviside's original motivation.
The coverup method can be used to make a partial fractions decomposition of a rational
function p)
whenever the denominator can be factored into distinct linearfactors.
We illustrate with an example; though simple, it should convince you that the method is
worth learning.
x7
Example
1.
Decompose
(
1)(
+2) into partial fractions.
Solution. We know the answer will have the form
x7
A
B
(x1)(x+2)
x1
+x+2
To determine
A
by
the coverup method, on the lefthand side we mentally remove (or cover
up with a finger) the factor x

1 associated with
A,
and substitute x = 1 into what's left;
this gives
A:
x7
17
(2)
((
+
2)x,=1
= 
=
2=
A.
1+2
Similarly,
B
is found by covering up the factor x
+
2 on the left, and substituting x = 2
into what's left. This gives
x7
27
x

= 3 =
B.
(x
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 Spring '11
 Blake
 Math, Calculus, Fraction, Elementary algebra, Rational function, Oliver Heaviside

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