# sn4_1 - Notes Week 4 Popper004 1 Calculate the given...

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Notes – Week 4 Popper004 1. Calculate the given integral: a) b) c) d) e) What about 3 sec xdx ?

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Section 8.5 – Partial Fraction Decomposition: Rational functions are defined as functions in the form ( ) ( ) ( ) P x F x G x = They are said to be proper if the degree of the numerator is less than the degree of the denominator (otherwise they are improper). Improper rational functions can be reduced using division as ( ) ( ) ( ) ( ) r x F x q x G x = + . Example1: Reduce by long division: 4 2 2 3 6 2 1 4 x x x x - + + + Methods to reduce rational functions (Partial fraction decomposition):

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We will start with functions whose denominators have linear factors. In general, each distinct linear factor ) ( a x - in the denominator gives rise to a term in the form: a x A - Example 2: 6 5 1 2 + + x x If the denominator has repeated linear factors , each factor of the form ( x α ) k in the denominator gives rise to
an expression of the form ( ) ( ) ( ) k k a x A a x A a x A - + + - + - ... 2

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sn4_1 - Notes Week 4 Popper004 1 Calculate the given...

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