Unformatted text preview: e bottom. What do you do if the denominator is not of lower degree than the numerator?
In a word: “DIVIDE” That is, you divide the denominator into the numerator using good old
fashioned Long Division  of course now we're dividing Polynomials and not numbers but it works
the same way, using “of lower degree than” instead of “less than”. 2 To integrate: a dx First write our fraction as a long division:
Now a goes into
, 2 times so we put 2 above the
then we multiply
so this goes under the
We subtract and get
which is of lower degree (in ) than
2
So
a 2 so it's the remainder
2 and our integral becomes NOW, we can do the second integral with partial fractions multiply by the denominator a a:
let now let : 2 and our integral now becomes ln
ln ln Homework:
Find the integrals
1) 4) 7) 2) 5) 8) 3) 6) 9) Likely answers: 1) ln
6) ln
; 7) ln ; 2) ln
; 8) ln ; 3) ln
; 9) ln ; 4) ln ; 5) ln ;...
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 Fall '13
 ProfessorBlank
 Calculus, Division, Fraction, Elementary arithmetic

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