Caluclus: Table of contents

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: \[\int\frac{x}{\left( x^2+b^2\right)^k}\,dx=\frac{\left( x^2+b^2\right)^{1-k}}{2 (1-k)} +C.\]

Practice problems

Problem 1. Evaluate the integral \(\displaystyle \int\frac1{x^2-9}\,dx\).

Problem 2. If the fraction \(\displaystyle \frac{7x^2 -2x +6}{(x -2)^2(x +3)}\) is expressed in the form \(\displaystyle \frac{\alpha}{x -2}+\frac{\beta}{(x-2)^2}+\frac{\gamma}{(x+3)}+\frac{\delta}{(x +3)^2}\), determine \(\alpha+\beta+\gamma+\delta\).

Problem 3. If the fraction \(\displaystyle \frac{7x^2 -9x +32}{\left(x^2+25\right)(x -3)}\) is expressed in the form \(\displaystyle \frac{\alpha x+\beta}{x^2+ 25}+\frac{\gamma}{(x-3)}\), determine \(\alpha-\beta+\gamma\).

Problem 4. Evaluate the integral \(\displaystyle \int_{100}^{200}\frac{ 4}{x^2 -7x +12 }\,dx\).

Problem 5. Evaluate the integral \(\displaystyle \int_0^1\frac{8x^2 +6x +5}{x^2 +2x+2 }\,dx\).