Integration of Rational Functions

Introduction

Rational functions are functions of the form \( f(x)=\frac{A(x)}{B(x)} \) where \( A \) and \( B \) are polynomials. Fortunately, all functions of this type can be integrated, and this section is devoted to building the techniques for their integration. The result of integration is always a function that has polynomials, logarithms, and inverse of \( \tan \) (denoted by \( \arctan \) in some countries and \( \tan^{-1} \) in others).

Partial fraction decomposition

Many integrals of fractions (such as \( \frac1{x^2-1} \)) can be evaluated by expressing the fraction as a sum of two simpler fractions. Let us first consider the following example:

Example 1
 
Evaluate the integral \( \displaystyle \int \frac{1}{x^2-1}\,dx \).

It turns out that any fraction of two polynomials can be represented as a sum of polynomials and terms of the form \( \displaystyle \frac1{(x+a)^k} \), \( \displaystyle \frac{1}{(x^2+b^2)^k} \), and \( \displaystyle \frac x{( x^2+b^2)^k} \). Let us first show by examples how this is done in practice, and in the end we present a theorem that guarantees such decomposition.

Example 2
 
Find real numbers \( a \), \( b \), and \( c \) such that \( \displaystyle f(x)=\frac{11x+15}{x(x-3)(x+5)}=\frac{a}{x}+\frac b{x-3}+\frac c{x+5} \), and then evaluate the integral \[ \int \frac{11x+15}{x(x-3)(x+5)}\,dx.\]

Example 3
 
Find a way to represent the fraction \( \displaystyle \frac{3x^2 +18x +29}{(x +2)^2(x +3)} \) in the form \( \displaystyle \frac{\alpha}{x +2}+\frac{\beta}{(x+2)^2}+\frac{\gamma}{(x+3)} \).

Example 4
 
Find the partial fractions expansion of \( \displaystyle \frac{3x^4-9x^3+20x^2-43x+14}{\left(x^2+4\right)(x -3)} \).

Example 5
 
Evaluate the integral \( \displaystyle \int\frac1{(x^2-a^2)^2}\,dx \).

Theorem 1
 
If \( A(x) \) and \( B(x) \) are polynomials such that \( B\neq 0 \), then there exist integers \( m \) and \( n \), a polynomial \( Q(x) \), sequences of real numbers \( (a_i)_{i=1}^m \), \( (b_i)_{i=1}^n \), \( (c_i)_{i=1}^n \), \( (d_i)_{i=1}^n \), \( (e_i)_{i=1}^n \), \( (f_i)_{i=1}^n \), and sequences of non-negative integers \( (k_i)_{i=1}^m \) and \( (l_i)_{i=1}^n \) such that \( b_i\neq 0 \) for all \( i\in\{1,2,\dots, n\} \) and: \[ \frac{A(x)}{B(x)}=Q(x)+\sum_{i=1}^m\frac{d_i}{\left(x+a_i\right)^{k_i}}+\sum_{i=1}^n\frac{e_ix+f_i}{\left((x+c_i)^2+b_i^2\right)^{l_i}}.\]

Integration of fractions of the form \( \displaystyle \frac1{(x+a)^k} \)

Theorem 2
 
If \( a \) is a real number then \[ \int \frac1{x+a}\,dx=\ln\left|x+a\right|+C,\] where \( C \) is any real number.

Theorem 3
 
If \( a \) is a real number and \( k \) an integer greater than \( 1 \), then \[ \int \frac1{(x+a)^k}\,dx =\frac{\left(x+a\right)^{1-k}}{1-k} +C.\]

Integration of fractions of the form \( \displaystyle \frac1{( x^2+b^2)^k} \)

Theorem 4
 
If \( b \) is a non-zero real number then \[ \int\frac1{ x^2+b^2}\,dx=\frac1{b}\tan^{-1}\left(\frac{x}b\right)+C.\]

In your first reading you may want to omit Theorems 5, 6, and 7 and skip to the next section.

Theorem 5
 
If \( b \) is a non-zero real number then \[ \int\frac1{\left( x^2+b^2\right)^2}\,dx= \frac1{2b^2}\left(\frac{x}{ x^2+b^2}+\frac1{b}\tan^{-1}\left(\frac{x}b\right) \right)+C.\]

Theorem 6
 
If \( b \) is a non-zero real number then \[ \int\frac1{\left( x^2+b^2\right)^3}\,dx=\frac{3 x^3+5b^2x}{8b^4\left( x^2+b^2\right)^2}+\frac{3\tan^{-1}\left(\frac{x}b\right)}{8b^5}+C. \]

The other integrals of the form \( \displaystyle \int\frac{1}{\left( x^2+b^2\right)^k}\,dx \) can be calculate in the same way by reducing to the integral with \( k-1 \).

Theorem 7
 
Assume that \( b \) is a non-zero real number and \( k> 1 \) a positive integer. Then \[ \int\frac1{\left( x^2+b^2\right)^k}\,dx= \frac1{2(k-1)b^2}\left(\frac{x}{\left( x^2+b^2\right)^{k-1}}+(2k-3)\int\frac1{\left( x^2+b^2\right)^{k-1}}\,dx \right).\]

Integration of fractions of the form \( \displaystyle \frac{x}{\left( x^2+b^2\right)^k} \)

Theorem 8
 
Assume that \( b \) is a non-zero real number. Then: \[ \int\frac{x}{ x^2+b^2}\,dx=\frac{\ln\left( x^2+b^2\right)}{2 } +C.\]

Theorem 9
 
Assume that \( k> 1 \) is a positive integer and \( b \) a non-zero real number. Then: \[ \int\frac{x}{\left( x^2+b^2\right)^k}\,dx=\frac{\left( x^2+b^2\right)^{1-k}}{2 (1-k)} +C.\]


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