Partial Fraction Decomposition Integrals of Partial Fractions

Lecture 10
Section 8.5 Rational Functions; Partial Fractions

Jiwen He

Department of Mathematics, University of Houston

jiwenhe@math.uh.edu
http://math.uh.edu/jiwenhe/Math1432
Z Z
A Bx + C
dx, dx
(x )k (x 2 + x + )k

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 1 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Rational Function
P(x)
Rational function: R(x) = Q(x) where P(x) and Q(x) are
polynomials.
2x 3x 4 20x 2 + 17
Yes: ,
x2 x 2 x 3 + 2x 2 7
1 x2 + 1
No: ,
x ln x
If degree(P) degree(Q), then, by division,
P(x) r (x)
= p(x) +
Q(x) Q(x)
r (x)
where p(x) is a polynomial and Q(x) is a proper rational
function (i.e., degree(r ) < degree(Q)).
x2 2x + 3
2
=1+ 2
x 2x 3 x 2x 3
x3 3x 2
2
=x +2+ 2
x 2x + 1 x 2x + 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 2 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Rational Function
P(x)
Rational function: R(x) = Q(x) where P(x) and Q(x) are
polynomials.
2x 3x 4 20x 2 + 17
Yes: ,
x2 x 2 x 3 + 2x 2 7
1 x2 + 1
No: ,
x ln x
If degree(P) degree(Q), then, by division,
P(x) r (x)
= p(x) +
Q(x) Q(x)
r (x)
where p(x) is a polynomial and Q(x) is a proper rational
function (i.e., degree(r ) < degree(Q)).
x2 2x + 3
2
=1+ 2
x 2x 3 x 2x 3
x3 3x 2
2
=x +2+ 2
x 2x + 1 x 2x + 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 2 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Rational Function
P(x)
Rational function: R(x) = Q(x) where P(x) and Q(x) are
polynomials.
2x 3x 4 20x 2 + 17
Yes: ,
x2 x 2 x 3 + 2x 2 7
1 x2 + 1
No: ,
x ln x
If degree(P) degree(Q), then, by division,
P(x) r (x)
= p(x) +
Q(x) Q(x)
r (x)
where p(x) is a polynomial and Q(x) is a proper rational
function (i.e., degree(r ) < degree(Q)).
x2 2x + 3
2
=1+ 2
x 2x 3 x 2x 3
x3 3x 2
2
=x +2+ 2
x 2x + 1 x 2x + 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 2 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Rational Function
P(x)
Rational function: R(x) = Q(x) where P(x) and Q(x) are
polynomials.
2x 3x 4 20x 2 + 17
Yes: ,
x2 x 2 x 3 + 2x 2 7
1 x2 + 1
No: ,
x ln x
If degree(P) degree(Q), then, by division,
P(x) r (x)
= p(x) +
Q(x) Q(x)
r (x)
where p(x) is a polynomial and Q(x) is a proper rational
function (i.e., degree(r ) < degree(Q)).
x2 2x + 3
2
=1+ 2
x 2x 3 x 2x 3
x3 3x 2
2
=x +2+ 2
x 2x + 1 x 2x + 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 2 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Rational Function
P(x)
Rational function: R(x) = Q(x) where P(x) and Q(x) are
polynomials.
2x 3x 4 20x 2 + 17
Yes: ,
x2 x 2 x 3 + 2x 2 7
1 x2 + 1
No: ,
x ln x
If degree(P) degree(Q), then, by division,
P(x) r (x)
= p(x) +
Q(x) Q(x)
r (x)
where p(x) is a polynomial and Q(x) is a proper rational
function (i.e., degree(r ) < degree(Q)).
x2 2x + 3
2
=1+ 2
x 2x 3 x 2x 3
x3 3x 2
2
=x +2+ 2
x 2x + 1 x 2x + 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 2 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Rational Function
P(x)
Rational function: R(x) = Q(x) where P(x) and Q(x) are
polynomials.
2x 3x 4 20x 2 + 17
Yes: ,
x2 x 2 x 3 + 2x 2 7
1 x2 + 1
No: ,
x ln x
If degree(P) degree(Q), then, by division,
P(x) r (x)
= p(x) +
Q(x) Q(x)
r (x)
where p(x) is a polynomial and Q(x) is a proper rational
function (i.e., degree(r ) < degree(Q)).
x2 2x + 3
2
=1+ 2
x 2x 3 x 2x 3
x3 3x 2
2
=x +2+ 2
x 2x + 1 x 2x + 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 2 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Rational Function
P(x)
Rational function: R(x) = Q(x) where P(x) and Q(x) are
polynomials.
2x 3x 4 20x 2 + 17
Yes: ,
x2 x 2 x 3 + 2x 2 7
1 x2 + 1
No: ,
x ln x
If degree(P) degree(Q), then, by division,
P(x) r (x)
= p(x) +
Q(x) Q(x)
r (x)
where p(x) is a polynomial and Q(x) is a proper rational
function (i.e., degree(r ) < degree(Q)).
x2 2x + 3
2
=1+ 2
x 2x 3 x 2x 3
x3 3x 2
2
=x +2+ 2
x 2x + 1 x 2x + 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 2 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition

Partial Fraction Decomposition
Let the denominator Q(x)
Y factor asY
Q(x) = a (x i )ni (x 2 + j x + j )nj
where the quadratic factors x 2 + j x + j are irreducible (i.e.,
j2 4j < 0, they have complex zeros).
P(x)
Proper rational function R(x) = Q(x) can be written as a sum
of partial fractions of the form:
A Bx + C
k
, :
(x ) (x + x + )k
2

Each power (x )n of a linear factor x contributes:

An A2 A1
n
+ 2
+
(x ) (x ) x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 3 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition

Partial Fraction Decomposition
Let the denominator Q(x)
Y factor asY
Q(x) = a (x i )ni (x 2 + j x + j )nj
where the quadratic factors x 2 + j x + j are irreducible (i.e.,
j2 4j < 0, they have complex zeros).
P(x)
Proper rational function R(x) = Q(x) can be written as a sum
of partial fractions of the form:
A Bx + C
k
, :
(x ) (x + x + )k
2

Each power (x )n of a linear factor x contributes:

An A2 A1
n
+ 2
+
(x ) (x ) x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 3 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition

Partial Fraction Decomposition
Let the denominator Q(x)
Y factor asY
Q(x) = a (x i )ni (x 2 + j x + j )nj
where the quadratic factors x 2 + j x + j are irreducible (i.e.,
j2 4j < 0, they have complex zeros).
P(x)
Proper rational function R(x) = Q(x) can be written as a sum
of partial fractions of the form:
A Bx + C
k
, :
(x ) (x + x + )k
2

Each power (x )n of a linear factor x contributes:

An A2 A1
n
+ 2
+
(x ) (x ) x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 3 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition

Partial Fraction Decomposition
Let the denominator Q(x)
Y factor asY
Q(x) = a (x i )ni (x 2 + j x + j )nj
where the quadratic factors x 2 + j x + j are irreducible (i.e.,
j2 4j < 0, they have complex zeros).
P(x)
Proper rational function R(x) = Q(x) can be written as a sum
of partial fractions of the form:
A Bx + C
k
, :
(x ) (x + x + )k
2

Each power (x )n of a linear factor x contributes:

An A2 A1
n
+ 2
+
(x ) (x ) x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 3 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
+ +
(x )n (x )2 x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
1 A B C Dx + E 1 1 x
= + 2+ + 2 = 3 + 2
x 3 (x 2 + 1) x 3 x x x +1 x x x +1

1 = (C + D)x 4 + (B + E )x 3 + (A + C )x 2 + Bx + A
A = 1, B = 0, A + C = 0, B + E = 0, C + D = 0
A = 1, B = 0, C = 1, E = 0, D = 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
+ +
(x )n (x )2 x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
1 A B C Dx + E 1 1 x
= + 2+ + 2 = 3 + 2
x 3 (x 2 + 1) x 3 x x x +1 x x x +1

1 = (C + D)x 4 + (B + E )x 3 + (A + C )x 2 + Bx + A
A = 1, B = 0, A + C = 0, B + E = 0, C + D = 0
A = 1, B = 0, C = 1, E = 0, D = 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
+ +
(x )n (x )2 x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
1 A B C Dx + E 1 1 x
= + 2+ + 2 = 3 + 2
x 3 (x 2 + 1) x 3 x x x +1 x x x +1

1 = (C + D)x 4 + (B + E )x 3 + (A + C )x 2 + Bx + A
A = 1, B = 0, A + C = 0, B + E = 0, C + D = 0
A = 1, B = 0, C = 1, E = 0, D = 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
+ +
(x )n (x )2 x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
1 A B C Dx + E 1 1 x
= + 2+ + 2 = 3 + 2
x 3 (x 2 + 1) x 3 x x x +1 x x x +1

1 = (C + D)x 4 + (B + E )x 3 + (A + C )x 2 + Bx + A
A = 1, B = 0, A + C = 0, B + E = 0, C + D = 0
A = 1, B = 0, C = 1, E = 0, D = 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
+ +
(x )n (x )2 x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
1 A B C Dx + E 1 1 x
= + 2+ + 2 = 3 + 2
x 3 (x 2 + 1) x 3 x x x +1 x x x +1

1 = (C + D)x 4 + (B + E )x 3 + (A + C )x 2 + Bx + A
A = 1, B = 0, A + C = 0, B + E = 0, C + D = 0
A = 1, B = 0, C = 1, E = 0, D = 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
+ +
(x )n (x )2 x
Each power (x 2 + x + )n of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
1 A B C Dx + E 1 1 x
= + 2+ + 2 = 3 + 2
x 3 (x 2 + 1) x 3 x x x +1 x x x +1

1 = (C + D)x 4 + (B + E )x 3 + (A + C )x 2 + Bx + A
A = 1, B = 0, A + C = 0, B + E = 0, C + D = 0
A = 1, B = 0, C = 1, E = 0, D = 1

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
+ 2 +
(x 2 + x + )n (x + x + )2 x 2 + x +
Examples
x2 A Bx + C Dx + E
2 2
= + 2 2
+ 2
(x 1)(x + 1) x 1 (x + 1) x +1
x 2 = (A + D)x 4 + (D + E )x 3 + (2A + B + D E )x 2
+ (B + C D + E )x + (A C E )
A + D = 0, D + E = 0, 2A + B + D E = 1,
B + C D + E = 0, A C E = 0 Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
+ 2 +
(x 2 + x + )n (x + x + )2 x 2 + x +
Examples
x2 A Bx + C Dx + E
2 2
= + 2 2
+ 2
(x 1)(x + 1) x 1 (x + 1) x +1
x 2 = (A + D)x 4 + (D + E )x 3 + (2A + B + D E )x 2
+ (B + C D + E )x + (A C E )
A + D = 0, D + E = 0, 2A + B + D E = 1,
B + C D + E = 0, A C E = 0 Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
+ 2 +
(x 2 + x + )n (x + x + )2 x 2 + x +
Examples
x2 A Bx + C Dx + E
2 2
= + 2 2
+ 2
(x 1)(x + 1) x 1 (x + 1) x +1
x 2 = (A + D)x 4 + (D + E )x 3 + (2A + B + D E )x 2
+ (B + C D + E )x + (A C E )
A + D = 0, D + E = 0, 2A + B + D E = 1,
B + C D + E = 0, A C E = 0 Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
+ 2 +
(x 2 + x + )n (x + x + )2 x 2 + x +
Examples
x2 A Bx + C Dx + E
2 2
= + 2 2
+ 2
(x 1)(x + 1) x 1 (x + 1) x +1
x 2 = (A + D)x 4 + (D + E )x 3 + (2A + B + D E )x 2
+ (B + C D + E )x + (A C E )
A + D = 0, D + E = 0, 2A + B + D E = 1,
B + C D + E = 0, A C E = 0 Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
+ 2 +
(x 2 + x + )n (x + x + )2 x 2 + x +
Examples
x2 A Bx + C Dx + E
2 2
= + 2 2
+ 2
(x 1)(x + 1) x 1 (x + 1) x +1
x 2 = (A + D)x 4 + (D + E )x 3 + (2A + B + D E )x 2
+ (B + C D + E )x + (A C E )
A + D = 0, D + E = 0, 2A + B + D E = 1,
B + C D + E = 0, A C E = 0 Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
x3 Ax + B Cx + D
2 2
= 2 2
+ 2
(x + 2x + 2) (x + 2x + 2) x + 2x + 2
x 3 = Cx 3 + (2C + D)x 2 + (A + 2C + 2D)x + (B + 2D)
C = 1, 2C + D = 0, A + 2C + 2D = 0, B + 2D = 0
Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
x3 Ax + B Cx + D
2 2
= 2 2
+ 2
(x + 2x + 2) (x + 2x + 2) x + 2x + 2
x 3 = Cx 3 + (2C + D)x 2 + (A + 2C + 2D)x + (B + 2D)
C = 1, 2C + D = 0, A + 2C + 2D = 0, B + 2D = 0
Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
x3 Ax + B Cx + D
2 2
= 2 2
+ 2
(x + 2x + 2) (x + 2x + 2) x + 2x + 2
x 3 = Cx 3 + (2C + D)x 2 + (A + 2C + 2D)x + (B + 2D)
C = 1, 2C + D = 0, A + 2C + 2D = 0, B + 2D = 0
Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
x3 Ax + B Cx + D
2 2
= 2 2
+ 2
(x + 2x + 2) (x + 2x + 2) x + 2x + 2
x 3 = Cx 3 + (2C + D)x 2 + (A + 2C + 2D)x + (B + 2D)
C = 1, 2C + D = 0, A + 2C + 2D = 0, B + 2D = 0
Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Rational Function Partial Fraction Decomposition

Partial Fraction Decomposition: Example

Each power (x )n of a linear factor x contributes:
An A2 A1
n
+ 2
+
(x ) (x ) x
2 n
Each power (x + x + ) of an irreducible quadratic factor
x 2 + x + contributes:
Bn x + Cn B2 x + C2 B1 x + C1
2 n
+ 2 2
+ 2
(x + x + ) (x + x + ) x + x +
Examples
x3 Ax + B Cx + D
2 2
= 2 2
+ 2
(x + 2x + 2) (x + 2x + 2) x + 2x + 2
x 3 = Cx 3 + (2C + D)x 2 + (A + 2C + 2D)x + (B + 2D)
C = 1, 2C + D = 0, A + 2C + 2D = 0, B + 2D = 0
Finish it

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 4 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
dx = +C
(x )k k 1 (x )k1

Examples
Z Z
6 6
3 2
dx = dx
x 5x + 6x x(x 2)(x 3)
Z  
1 3 2
= + dx
x x 2 x 3
= ln |x| 3 ln |x 2| + 2 ln |x 3| + C

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
dx = +C
(x )k k 1 (x )k1

Examples
Z Z
6 6
3 2
dx = dx
x 5x + 6x x(x 2)(x 3)
Z  
1 3 2
= + dx
x x 2 x 3
= ln |x| 3 ln |x 2| + 2 ln |x 3| + C

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
dx = +C
(x )k k 1 (x )k1

Examples
Z Z
6 6
3 2
dx = dx
x 5x + 6x x(x 2)(x 3)
Z  
1 3 2
= + dx
x x 2 x 3
= ln |x| 3 ln |x 2| + 2 ln |x 3| + C

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
dx = +C
(x )k k 1 (x )k1

Examples
Z Z
6 6
3 2
dx = dx
x 5x + 6x x(x 2)(x 3)
Z  
1 3 2
= + dx
x x 2 x 3
= ln |x| 3 ln |x 2| + 2 ln |x 3| + C

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples
Z Z
1 1
3 2
dx = 2
dx
x 2x x (x 2)
Z  
1/2 1/4 1/4
= + + dx
x2 x x 2
 
1 2
= ln |x| + ln |x 2| + C
4 x

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples
Z Z
1 1
3 2
dx = 2
dx
x 2x x (x 2)
Z  
1/2 1/4 1/4
= + + dx
x2 x x 2
 
1 2
= ln |x| + ln |x 2| + C
4 x

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples
Z Z
1 1
3 2
dx = 2
dx
x 2x x (x 2)
Z  
1/2 1/4 1/4
= + + dx
x2 x x 2
 
1 2
= ln |x| + ln |x 2| + C
4 x

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples
Z Z
1 1
3 2
dx = 2
dx
x 2x x (x 2)
Z  
1/2 1/4 1/4
= + + dx
x2 x x 2
 
1 2
= ln |x| + ln |x 2| + C
4 x

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples
x2 x2
Z Z
dx = dx
(x 2 9)2 (x 3)2 (x + 3)2
 
1/12
Z
1/4 1/12 1/4
= + + + dx
(x 3)2 x 3 (x + 3)2 x +3
 
1 3 3
= + ln |x 3| ln |x + 3| + C
12 x 3 x +3

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples
x2 x2
Z Z
dx = dx
(x 2 9)2 (x 3)2 (x + 3)2
 
1/12
Z
1/4 1/12 1/4
= + + + dx
(x 3)2 x 3 (x + 3)2 x +3
 
1 3 3
= + ln |x 3| ln |x + 3| + C
12 x 3 x +3

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples
x2 x2
Z Z
dx = dx
(x 2 9)2 (x 3)2 (x + 3)2
 
1/12
Z
1/4 1/12 1/4
= + + + dx
(x 3)2 x 3 (x + 3)2 x +3
 
1 3 3
= + ln |x 3| ln |x + 3| + C
12 x 3 x +3

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

A
Partial Fractions: (x)k
Z
A
dx = A ln |x | + C
x
Z
A A 1
k
dx = +C
(x ) k 1 (x )k1

Examples
x2 x2
Z Z
dx = dx
(x 2 9)2 (x 3)2 (x + 3)2
 
1/12
Z
1/4 1/12 1/4
= + + + dx
(x 3)2 x 3 (x + 3)2 x +3
 
1 3 3
= + ln |x 3| ln |x + 3| + C
12 x 3 x +3

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 5 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z
2x + 1
du = ln |u| + C = ln x 2 + x +  + C
 
2
dx =
x + x + u
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z
2x + 1
du = ln |u| + C = ln x 2 + x +  + C
 
2
dx =
x + x + u
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z
2x + 1
du = ln |u| + C = ln x 2 + x +  + C
 
2
dx =
x + x + u
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z
2x + 1
du = ln |u| + C = ln x 2 + x +  + C
 
2
dx =
x + x + u
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z
2x + 1
du = ln |u| + C = ln x 2 + x +  + C
 
2
dx =
x + x + u
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z
2x + 1
du = ln |u| + C = ln x 2 + x +  + C
 
2
dx =
x + x + u
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z
2x + 1
du = ln |u| + C = ln x 2 + x +  + C
 
2
dx =
x + x + u
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z
2x + 1
du = ln |u| + C = ln x 2 + x +  + C
 
2
dx =
x + x + u
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z Z
1 1 1
dx = dt = a sec2 u du
x 2 + x + t 2 + a2 a2 sec2 u
x + 2
Z
1 1 1 t 1
= du = u + C = tan1 = q tan1 q
a a a a 2 2
4 4

Note x 2 + x + = (x + /2)2 + 2 /4.

Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z Z
1 1 1
dx = dt = a sec2 u du
x 2 + x + t 2 + a2 a2 sec2 u
x + 2
Z
1 1 1 t 1
= du = u + C = tan1 = q tan1 q
a a a a 2 2
4 4

Note x 2 + x + = (x + /2)2 + 2 /4.

Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z Z
1 1 1
dx = dt = a sec2 u du
x 2 + x + t 2 + a2 a2 sec2 u
x + 2
Z
1 1 1 t 1
= du = u + C = tan1 = q tan1 q
a a a a 2 2
4 4

Note x 2 + x + = (x + /2)2 + 2 /4.

Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z Z
1 1 1
dx = dt = a sec2 u du
x 2 + x + t 2 + a2 a2 sec2 u
x + 2
Z
1 1 1 t 1
= du = u + C = tan1 = q tan1 q
a a a a 2 2
4 4

Note x 2 + x + = (x + /2)2 + 2 /4.

Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z Z
1 1 1
dx = dt = a sec2 u du
x 2 + x + t 2 + a2 a2 sec2 u
x + 2
Z
1 1 1 t 1
= du = u + C = tan1 = q tan1 q
a a a a 2 2
4 4

Note x 2 + x + = (x + /2)2 + 2 /4.

Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z Z
1 1 1
dx = dt = a sec2 u du
x 2 + x + t 2 + a2 a2 sec2 u
x + 2
Z
1 1 1 t 1
= du = u + C = tan1 = q tan1 q
a a a a 2 2
4 4

Note x 2 + x + = (x + /2)2 + 2 /4.

Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z Z
1 1 1
dx = dt = a sec2 u du
x 2 + x + t 2 + a2 a2 sec2 u
x + 2
Z
1 1 1 t 1
= du = u + C = tan1 = q tan1 q
a a a a 2 2
4 4

Note x 2 + x + = (x + /2)2 + 2 /4.

Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

C B2 1 x + 2
Z
Bx + C B  2 
dx = ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Proof.
Z Z Z
1 1 1
dx = dt = a sec2 u du
x 2 + x + t 2 + a2 a2 sec2 u
x + 2
Z
1 1 1 t 1
= du = u + C = tan1 = q tan1 q
a a a a 2 2
4 4

Note x 2 + x + = (x + /2)2 + 2 /4.

Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 6 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
x2
Z  
4x 4
Z
1 1
dx = + dx
(x + 1)(x 2 + 4) 5 x + 1 x2 + 4
Z  
1 1 4x 4
= + dx
5 x + 1 x2 + 4 x2 + 4
1 x
ln |x + 1| + 2 ln x 2 + 4 2 tan1


= +C
5 2

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
x2
Z  
4x 4
Z
1 1
dx = + dx
(x + 1)(x 2 + 4) 5 x + 1 x2 + 4
Z  
1 1 4x 4
= + dx
5 x + 1 x2 + 4 x2 + 4
1 x
ln |x + 1| + 2 ln x 2 + 4 2 tan1


= +C
5 2

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
x2
Z  
4x 4
Z
1 1
dx = + dx
(x + 1)(x 2 + 4) 5 x + 1 x2 + 4
Z  
1 1 4x 4
= + dx
5 x + 1 x2 + 4 x2 + 4
1 x
ln |x + 1| + 2 ln x 2 + 4 2 tan1


= +C
5 2

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
x2
Z  
4x 4
Z
1 1
dx = + dx
(x + 1)(x 2 + 4) 5 x + 1 x2 + 4
Z  
1 1 4x 4
= + dx
5 x + 1 x2 + 4 x2 + 4
1 x
ln |x + 1| + 2 ln x 2 + 4 2 tan1


= +C
5 2

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + x 2 +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
x2
Z  
4x 4
Z
1 1
dx = + dx
(x + 1)(x 2 + 4) 5 x + 1 x2 + 4
Z  
1 1 4x 4
= + dx
5 x + 1 x2 + 4 x2 + 4
1 x
ln |x + 1| + 2 ln x 2 + 4 2 tan1


= +C
5 2

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
x 2 + 5x + 2
Z  
1
Z
2x + 3
dx = + dx
(x + 1)(x 2 + 1) x + 1 x2 + 1
Z  
1 2x 3
= + + dx
x + 1 x2 + 1 x2 + 1
= ln |x + 1| + ln x 2 + 1 + 3 tan1 x + C

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
x 2 + 5x + 2
Z  
1
Z
2x + 3
dx = + dx
(x + 1)(x 2 + 1) x + 1 x2 + 1
Z  
1 2x 3
= + + dx
x + 1 x2 + 1 x2 + 1
= ln |x + 1| + ln x 2 + 1 + 3 tan1 x + C

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
x 2 + 5x + 2
Z  
1
Z
2x + 3
dx = + dx
(x + 1)(x 2 + 1) x + 1 x2 + 1
Z  
1 2x 3
= + + dx
x + 1 x2 + 1 x2 + 1
= ln |x + 1| + ln x 2 + 1 + 3 tan1 x + C

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
x 2 + 5x + 2
Z  
1
Z
2x + 3
dx = + dx
(x + 1)(x 2 + 1) x + 1 x2 + 1
Z  
1 2x 3
= + + dx
x + 1 x2 + 1 x2 + 1
= ln |x + 1| + ln x 2 + 1 + 3 tan1 x + C

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
Z  
x 1
Z
1 1
dx = + 2 dx
x(x 2 + x + 1) x x +x +1
Z  
1 1 2x 1 1
= dx
x 2 x2 + x + 1 2 x2 + x + 1
  
1 2

1 1 2 1
= ln |x| ln x + x + 1 tan x+ +C
2 3 3 2

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
Z  
x 1
Z
1 1
dx = + 2 dx
x(x 2 + x + 1) x x +x +1
Z  
1 1 2x 1 1
= dx
x 2 x2 + x + 1 2 x2 + x + 1
  
1 2

1 1 2 1
= ln |x| ln x + x + 1 tan x+ +C
2 3 3 2

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
Z  
x 1
Z
1 1
dx = + 2 dx
x(x 2 + x + 1) x x +x +1
Z  
1 1 2x 1 1
= dx
x 2 x2 + x + 1 2 x2 + x + 1
  
1 2

1 1 2 1
= ln |x| ln x + x + 1 tan x+ +C
2 3 3 2

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: x 2 +x+
= 2 x 2 +x+ + 2
x +x+

 C B2
1 x + 2
Z
Bx + C B  2
dx= ln x + x +  +q tan
x 2 + x +
q
2 2 2
4 4

Example
Z  
x 1
Z
1 1
dx = + 2 dx
x(x 2 + x + 1) x x +x +1
Z  
1 1 2x 1 1
= dx
x 2 x2 + x + 1 2 x2 + x + 1
  
1 2

1 1 2 1
= ln |x| ln x + x + 1 tan x+ +C
2 3 3 2

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 7 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z
2x + 1 1 1
dx = du = +C
(x 2 + x + )k uk k 1 u k1
1 1
= +C
k 1 (x 2 + x + )k1
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z
2x + 1 1 1
dx = du = +C
(x 2 + x + )k uk k 1 u k1
1 1
= +C
k 1 (x 2 + x + )k1
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z
2x + 1 1 1
dx = du = +C
(x 2 + x + )k uk k 1 u k1
1 1
= +C
k 1 (x 2 + x + )k1
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z
2x + 1 1 1
dx = du = +C
(x 2 + x + )k uk k 1 u k1
1 1
= +C
k 1 (x 2 + x + )k1
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z
2x + 1 1 1
dx = du = +C
(x 2 + x + )k uk k 1 u k1
1 1
= +C
k 1 (x 2 + x + )k1
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z
2x + 1 1 1
dx = du = +C
(x 2 + x + )k uk k 1 u k1
1 1
= +C
k 1 (x 2 + x + )k1
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z
2x + 1 1 1
dx = du = +C
(x 2 + x + )k uk k 1 u k1
1 1
= +C
k 1 (x 2 + x + )k1
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z
2x + 1 1 1
dx = du = +C
(x 2 + x + )k uk k 1 u k1
1 1
= +C
k 1 (x 2 + x + )k1
Set u = x 2 + x + , du = 2x + dx.

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x +x+)k
2

Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z Z
1 1 1
2 k
dx = 2 2 k
dt = a sec2 u du
(x + x + ) (t + a ) (a sec2 u)k
2
Z Z
1 1 1
= 2k1 du = 2k1 cos2(k1) u du =
a sec2(k1) u a
Note x 2 + x + = (x + /2)2 + 2 /4.
Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
n1
Z Z
n 1 n1
Reduction: cos x dx = cos x sin x + cosn2 x dx
n n

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x +x+)k
2

Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z Z
1 1 1
2 k
dx = 2 2 k
dt = a sec2 u du
(x + x + ) (t + a ) (a sec2 u)k
2
Z Z
1 1 1
= 2k1 du = 2k1 cos2(k1) u du =
a sec2(k1) u a
Note x 2 + x + = (x + /2)2 + 2 /4.
Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
n1
Z Z
n 1 n1
Reduction: cos x dx = cos x sin x + cosn2 x dx
n n

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x +x+)k
2

Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z Z
1 1 1
2 k
dx = 2 2 k
dt = a sec2 u du
(x + x + ) (t + a ) (a sec2 u)k
2
Z Z
1 1 1
= 2k1 du = 2k1 cos2(k1) u du =
a sec2(k1) u a
Note x 2 + x + = (x + /2)2 + 2 /4.
Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
n1
Z Z
n 1 n1
Reduction: cos x dx = cos x sin x + cosn2 x dx
n n

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x +x+)k
2

Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z Z
1 1 1
2 k
dx = 2 2 k
dt = a sec2 u du
(x + x + ) (t + a ) (a sec2 u)k
2
Z Z
1 1 1
= 2k1 du = 2k1 cos2(k1) u du =
a sec2(k1) u a
Note x 2 + x + = (x + /2)2 + 2 /4.
Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
n1
Z Z
n 1 n1
Reduction: cos x dx = cos x sin x + cosn2 x dx
n n

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x +x+)k
2

Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z Z
1 1 1
2 k
dx = 2 2 k
dt = a sec2 u du
(x + x + ) (t + a ) (a sec2 u)k
2
Z Z
1 1 1
= 2k1 du = 2k1 cos2(k1) u du =
a sec2(k1) u a
Note x 2 + x + = (x + /2)2 + 2 /4.
Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
n1
Z Z
n 1 n1
Reduction: cos x dx = cos x sin x + cosn2 x dx
n n

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x +x+)k
2

Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z Z
1 1 1
2 k
dx = 2 2 k
dt = a sec2 u du
(x + x + ) (t + a ) (a sec2 u)k
2
Z Z
1 1 1
= 2k1 du = 2k1 cos2(k1) u du =
a sec2(k1) u a
Note x 2 + x + = (x + /2)2 + 2 /4.
Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
n1
Z Z
n 1 n1
Reduction: cos x dx = cos x sin x + cosn2 x dx
n n

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x +x+)k
2

Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z Z
1 1 1
2 k
dx = 2 2 k
dt = a sec2 u du
(x + x + ) (t + a ) (a sec2 u)k
2
Z Z
1 1 1
= 2k1 du = 2k1 cos2(k1) u du =
a sec2(k1) u a
Note x 2 + x + = (x + /2)2 + 2 /4.
Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
n1
Z Z
n 1 n1
Reduction: cos x dx = cos x sin x + cosn2 x dx
n n

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x +x+)k
2

Z Z
Bx + C B 1
2 k
dx = +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

Proof.
Z Z Z
1 1 1
2 k
dx = 2 2 k
dt = a sec2 u du
(x + x + ) (t + a ) (a sec2 u)k
2
Z Z
1 1 1
= 2k1 du = 2k1 cos2(k1) u du =
a sec2(k1) u a
Note x 2 + x + = (x + /2)2 + 2 /4.
Set t = x + /2, a2 = 2 /4.
set a tan u = t, a sec2 u du = dt, t 2 + a2 = a2 sec2 u.
n1
Z Z
n 1 n1
Reduction: cos x dx = cos x sin x + cosn2 x dx
n n

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 8 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx= +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

C B2
where c = , t = x + /2, a2 = 2 /4, a tan u = t.
a2k1
Example
Z 4
3x + x 3 + 20x 2 + 3x + 31
Z 
2 x 1
dx = + dx
(x + 1)(x 2 + 4)2 x + 1 x 2 + 4 (x 2 + 4)2
Z
1 2

1
= 2 ln |x + 1| + ln x + 4 cos2 u du
2 8
1 1
= 2 ln |x + 1| + ln x 2 + 4 (u + sin u cos u) + C


2 16  
1 2

1 1 x 2x
= 2 ln |x + 1| + ln x + 4 tan + +C
2 16 2 x2 + 4
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 9 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx= +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

C B2
where c = , t = x + /2, a2 = 2 /4, a tan u = t.
a2k1
Example
Z 4
3x + x 3 + 20x 2 + 3x + 31
Z 
2 x 1
dx = + dx
(x + 1)(x 2 + 4)2 x + 1 x 2 + 4 (x 2 + 4)2
Z
1 2

1
= 2 ln |x + 1| + ln x + 4 cos2 u du
2 8
1 1
= 2 ln |x + 1| + ln x 2 + 4 (u + sin u cos u) + C


2 16  
1 2

1 1 x 2x
= 2 ln |x + 1| + ln x + 4 tan + +C
2 16 2 x2 + 4
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 9 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx= +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

C B2
where c = , t = x + /2, a2 = 2 /4, a tan u = t.
a2k1
Example
Z 4
3x + x 3 + 20x 2 + 3x + 31
Z 
2 x 1
dx = + dx
(x + 1)(x 2 + 4)2 x + 1 x 2 + 4 (x 2 + 4)2
Z
1 2

1
= 2 ln |x + 1| + ln x + 4 cos2 u du
2 8
1 1
= 2 ln |x + 1| + ln x 2 + 4 (u + sin u cos u) + C


2 16  
1 2

1 1 x 2x
= 2 ln |x + 1| + ln x + 4 tan + +C
2 16 2 x2 + 4
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 9 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx= +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

C B2
where c = , t = x + /2, a2 = 2 /4, a tan u = t.
a2k1
Example
Z 4
3x + x 3 + 20x 2 + 3x + 31
Z 
2 x 1
dx = + dx
(x + 1)(x 2 + 4)2 x + 1 x 2 + 4 (x 2 + 4)2
Z
1 2

1
= 2 ln |x + 1| + ln x + 4 cos2 u du
2 8
1 1
= 2 ln |x + 1| + ln x 2 + 4 (u + sin u cos u) + C


2 16  
1 2

1 1 x 2x
= 2 ln |x + 1| + ln x + 4 tan + +C
2 16 2 x2 + 4
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 9 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx= +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

C B2
where c = , t = x + /2, a2 = 2 /4, a tan u = t.
a2k1
Example
Z 4
3x + x 3 + 20x 2 + 3x + 31
Z 
2 x 1
dx = + dx
(x + 1)(x 2 + 4)2 x + 1 x 2 + 4 (x 2 + 4)2
Z
1 2

1
= 2 ln |x + 1| + ln x + 4 cos2 u du
2 8
1 1
= 2 ln |x + 1| + ln x 2 + 4 (u + sin u cos u) + C


2 16  
1 2

1 1 x 2x
= 2 ln |x + 1| + ln x + 4 tan + +C
2 16 2 x2 + 4
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 9 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Integrals of Partial Fractions: Examples

Bx+C B 2x+ C B2
Partial Fractions: (x 2 +x+)k
= 2 (x 2 +x+)k + (x 2 +x+)k
Z Z
Bx + C B 1
2 k
dx= +c cos2(k1) u du
(x + x + ) 2(k 1) (x + x + )k1
2

C B2
where c = , t = x + /2, a2 = 2 /4, a tan u = t.
a2k1
Example
Z 4
3x + x 3 + 20x 2 + 3x + 31
Z 
2 x 1
dx = + dx
(x + 1)(x 2 + 4)2 x + 1 x 2 + 4 (x 2 + 4)2
Z
1 2

1
= 2 ln |x + 1| + ln x + 4 cos2 u du
2 8
1 1
= 2 ln |x + 1| + ln x 2 + 4 (u + sin u cos u) + C


2 16  
1 2

1 1 x 2x
= 2 ln |x + 1| + ln x + 4 tan + +C
2 16 2 x2 + 4
Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 9 / 10
Partial Fraction Decomposition Integrals of Partial Fractions Partial Fractions

Outline

Partial Fraction Decomposition

Rational Function
Partial Fraction Decomposition

Integrals of Partial Fractions

Partial Fractions

Jiwen He, University of Houston Math 1432 Section 26626, Lecture 10 February 14, 2008 10 / 10