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Unformatted text preview: and the identity .
Then , and
Alternatively, you can use and the identity . Then
, and
Note: Although we have very speciﬁc bounds, their choice is not very important: one could choose
for the restriction when dealing with , for example. We need only ensure that the func
tion is onetoone. Furthermore, we could have used , , and instead of , , and
respectively, since they have the same desired properties, but they would work on a different interval. For
example, is not onetoone for , but is onetoone for .
As always with substitution, emember to change the bounds of integration. If it is an indeﬁnite integral,
remember to return to the original variable. However, be careful  since we used an inverse substitution, the
old variable is a function of the new one rather than the other way around. When dealing with trigonometric substitution, we often end up with a trigonometric term such as when we began with a substitution
like . Rather than use , you will be required to simplify the expression into some
thing that requires no trigonometric functions. We do this by using the information garnered from the
righttriangle produced from our substitution equation.
5 WATERLOO SOS E XAM AID: MATH138 M IDTERM Partial Fractions:
When confronted with a rational function (a ratio of polynomials), we can try using partial fractions in a
somewhat algorithmic way  as long as we can factor polynomials, divide polynomials, and solve systems
of equations. Suppose you must ﬁnd , for , where and are polynomials in
. We will apply the following algorithm to express the solution as a sum of rational functions, lograrithms
of polynomials, and arctan of polynomials.
Step 1: If necessary, use polynomial division to express
are polynomials with .
, where and Step 2: Completely factor the denominator . A theorem from algebra guarantees a factorization
into linear terms (of the form ) and irreducible quadratic terms (of the form
such that ). Common tricks to use here are common factors, difference of squares,
sum and difference of cubes, quadratic formula, and temporarily considering as the variable
instead of .
Step 3: Express
as a sum of partial fractions of the form
or where and appears as a factor of , or and
appears as a factor of . To be explicit:
(i) If has a linear term with multiplicity , say (and ),
then add
(ii) If has a quadratic term with multiplicity , say (and
), then add
We can solve for the coefﬁcients by clearing the denominators using crossmultiplication, then
compa...
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This document was uploaded on 03/04/2014 for the course MATH 138 at Waterloo.
 Winter '07
 Anoymous
 Math, Calculus

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