7'4 Integration of Rational Functions by Partial Fractions

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7.4 Integration of Rational Functions by Partial
Fractions

• This section examines a procedure for
  decomposing a rational function into simpler
  rational functions to which we can apply the
  basic integration formulas.
• Partial Fraction Decomposition of
• Refer to the Handout of Integration of
  Rational Functions by Partial Fractions
• Rationalizing Substitutions
• Rational Functions of Sine and Cosine
  (Weierstrass Substitutions)

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A. Examples of Integrals of Rational Functions
Distinct Linear Factors
Solution
Note If the degree of the numerator is greater
than or equal to the degree of the denominator,
perform the long division first.
where,
To find the values of A and B, multiply both
sides by the lowest common denominator which
leads to the basic equation.
Basic equation
To solve for A, let in basic
equation. We chose to
eliminate the term
Similary, to solve for B, let
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Repeated Linear Factors
Solution
We include one fraction for each power of x and
(x 1)
Basic Equation
let
let
Note Since we have exhausted the most
convenient choices for x, to find the value of B,
use any other value of x along with the
calculated values of A and C
let
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Distinct Irreducible Quadratic Factor
Solution
The fraction with quadratic factor assigns two
contants to be determined with the first constant
multiplied by x
Basic Equation

• Note If the decomposition involves a quadratic
  factor, an alternate procedure is more
  convenient as outlined
• Expand the basic equation.
• Collect terms according to powers of x.
• Equate the coefficients of like powers to obtain
  a system of linear equations involving A, B, C,
  and so on.
• Solve the system of linear equations.

Remark
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Repeated Irreducible Quadratic Factors
Solution
Basic Equation
Equating Coefficients
Integrate this by Trigonometric Substitution.
See prior section.