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Summary Partial Fraction Decomposition | Calculus II Notes CA$12.25
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Summary Partial Fraction Decomposition | Calculus II Notes
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Details how to integrate some rational functions using partial decomposition. Includes broken down and step by step instructions to solve examples for linear factor cases and irreducible quadratic factor in the denominator.

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  • July 16, 2023
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  • 2022/2023
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1.10 Partial Fraction Decomposition
for integrating some rational functions


First:if the degree of the numerator is greater than the degree of the denominator, do long division



eg.( xs,2432dx
x3 -

3x quotient
x
2
3x5
+ -
6x + 2 divisor numerator
u
=(Xs 3x") +




x(x3) -

343 -

6x 2 +




-- (3x 9x) -




x(3x) 3x+2 divisor, stop
when the
degree of the remainder is less than the long division

numerator quotient
- remainder
+




divisor divisor



x5 6x 2 (x3 3x)
3
- +
- +




2*3*
=




23
x +
3
e
S x5 -6x 2dx f(x3 x
2
+

3
+
=
-
3x)dx 3x
+



S 2
x
+




3
+




=** 2
3)Y
+
+




un
ux+
=
3
Y5
u =




du 2XdX
=




du
dx
=




I(=en(u) 3( - arctan(u)


arctan(*)
=** x en(x+3) +o
- + +




3



you only need to look at
see
when
degreep(x) degree q(x)
"Partial fractions"


eg. S, dx factor the denominator ( (x2-1) (x 1)(x 1) =
- +




write
integral as sum offrations
1 = A B
+



-
x 1 x 1 +
x -
1


A(X-1) B(x+1)
= +




x2 -
1

solve for numerators
a) equatenumerators after combining fractions
1 A(x-1) B(x+ 1)
= +



expand, collectlike terms, factor out common is
0x + 1 (A B)x=
+
(
+ -

A B)+
-




Ax Bx +
0x
=




-
A B
+
1
=

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