c Dr Oksana Shatalov, Fall 2012
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8.4: Integration Of Rational Functions By Partial Fractions
EXAMPLE 1. Evaluate the following integrals:
Z
2x − 5
(a)
dx
2
x − 5x + 4
x−6
dx
− 5x + 4
We will use so called Partial Fraction Decomposition
Z
(b)
x2
c Dr Oksana Shatalov, Fall 2012
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Partial Fraction Decomposition Process
P (x)
Rational function: f (x) =
, where P (x) and Q(x) are polynomials.
Q(x)
Step 0. deg P (x) < deg Q(x).
Important to remember: Partial fractions can only be done if the degree of the numerator is
strictly less than the degree of denominator. (Otherwise, you must first do long division.)
Step 1. Factor the denominator as much as possible.
Step 2. For each linear factor in the denominator use the following table:
Factor in denominator Term in partial fraction decomposition
ax + b
A
ax + b
(ax + b)2
A2
A1
+
ax + b (ax + b)2
(ax + b)k
A1
A2
Ak
+
+ ··· +
2
ax + b (ax + b)
(ax + b)k
For each irreducible quadratic factor in the denominator use the following table:
Factor in denominator Term in partial fraction decomposition
ax2 + bx + c
Ax + B
ax2 + bx + c
(ax2 + bx + c)2
A1 x + B1
A2 x + B2
+
2
ax + bx + c (ax2 + bx + c)2
(ax2 + bx + c)k
A1 x + B1
A2 x + B2
Ak x + Bk
+ ··· +
+
2
2
2
ax + bx + c (ax + bx + c)
(ax2 + bx + c)k
Z
EXAMPLE 2. Evaluate I =
3x + 11
dx
−x−6
x2
c Dr Oksana Shatalov, Fall 2012
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Z
EXAMPLE 3. Evaluate I =
x2 + 1
dx
x2 − x
c Dr Oksana Shatalov, Fall 2012
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Z
EXAMPLE 4. Evaluate I =
x2
dx
(x − 3)(x + 2)2
c Dr Oksana Shatalov, Fall 2012
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Z
EXAMPLE 5. Integrate I =
x4 − x3 − 12x2 + 10
dx
x3 − 4x2
c Dr Oksana Shatalov, Fall 2012
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EXAMPLE 6. Write out the form of the partial fraction decomposition of the following function:
x3 + 5x2 − 2012
f (x) =
x(x + 12)(x2 + 2x − 3)(x2 + x + 1)(x2 + 25)3