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
=
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
=
