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 $\int\frac1{ x^2+b^2}\,dx=\frac1{b}\tan^{-1}\left(\frac{x}b\right)+C.$

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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