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Integration by partial fractions To integrate the fraction

one cannot use all the previous methods (substitution, by parts).

However, one observe that the fraction can be reduced into

i.e. it is the sum of other fractions. Then

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In this section we study the technique to integrate fractions

We will see that the fraction

can be reduced into sum (or difference) of other fractions called partial fractions.

This is called the partial fractions expansion of .

Then the integral

is the sum

of the integrals of partial fractions.

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EX

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If we factorize

we call

In the case (or ), the term is called simple.

If (or ), the term is said to be repeated with order (resp., ).

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EX The polynomial

has a simple linear term , a repeated linear term , order a repeated quad term ,

order 2

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Partial Fraction (with Long division)

If , using long division we can express

where . So

can be easily integrated.

If then . So long division is not required.

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EX Use long division to reduce

Then integrate

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Now we have to study

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Partial Fraction: (simple linear terms) If

where are distinct real numbers, then there are such that

can be computed from

Formula

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EX Find the partial fraction expansion of

Then integrate

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EX Find the partial fraction expansion of

Then evaluate

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Generally, if the factorization of has a simple linear term (together with others, which could be repeated, or quadratics), this term gives rise to

in the partial fraction expansion of

is computed by the same formula:

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Partial Fraction: (repeated linear term)

If the factorization of has

then this term gives rise to

in the partial fraction expansion of

.

can be computed from

Formula

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EX Find the partial fraction expansion of

Then evaluate the integral

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EX Evaluate

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Partial Fraction: (simple quad terms)

If the factorization of has a simple quad term

then this term gives rise to

in the partial fraction expansion of

.

Formula

Repeated quad terms need the trig. subst., will be discussed later.

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EX Find the partial fraction expansion of

Then evaluate the integral

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EX (Change of variable first) Evaluate

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EX (Change of variable first) Evaluate