Math 176 Calculus – Sec. 7.4: Integration of Rational Functions by Partial Fractions
I. Reducing an Improper Fraction (long division)
A. Introduction When the degree of the numerator is greater than or equal to the degree of the
denominator, we will be using long division to simplify the integrand. Sometimes after doing long
division, you will have a remainder that will be placed on top of the divisor; this part of the new
integrand may result after integration in either a natural logarithm or an arctangent.
∫x
2
dx
1
⎛ x⎞
= tan −1 ⎜ ⎟ + C
2
a
+a
⎝a⎠
B. Examples
1.
2.
⎛ x2 + 7x − 3 ⎞
∫ ⎜⎜⎝ x + 4 ⎟⎟⎠dx
⎛ 3 x 5 + 28 x ⎞
∫ ⎜⎜⎝ x 4 + 9 ⎟⎟⎠dx
 4x 2 − 7 
 dx
3. ∫ 

2x
+
3

−1
3
II.
Partial Fractions
A. Introduction
Partial Fractions consists of decomposing a rational function into simpler component
fractions and then evaluating the integral term by term.
B. Denominator is a Product of Distinct Linear Factors
∫x
2
3x + 7
dx
+ 6x + 5
C. Denominator is a Product of Linear Factors, Some of Which are Repeated
∫
3x 2 − 8x + 13
( x + 3)( x − 1)
2
dx
D. Denominator Contains Irreducible Quadratic Factors, None of Which is Repeated
2x 2 + x − 8
∫ x 3 + 4x dx
I I I . Rationalizing Substitutions
Some nonrational functions can be changed into rational functions by means of appropriate
substitutions.
x
∫ x −4
IV.
dx
Additional Examples
Evaluate the following:
1.
∫x
2
2x + 1
dx
+ 2x − 3
2.
3.
∫(
x 2 - x − 21
dx
x 2 + 4 (2x − 1)
)
4x 2 + 3x + 6
∫ x 2 x 2 + 3 dx
(
)
ey
4.
∫
5.
∫ (1 + e )
6.
16 − e2 y
dy
dx
x
∫(
e2t
dt
et − 2
)
7.
∫
x 2 + x + 1 dx