Rational Function Integration | Complete Guide with Examples

What You'll Learn

• Standard method for rational function integration using partial fractions
• How to decompose complex rational functions
• Special techniques for specific rational function forms
• Multiple worked examples with detailed solutions

1. Fundamentals

Definition 1.1: Rational Function

A rational function is a function of the form:

R(x) = \frac{P(x)}{Q(x)}

where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$ .

Definition 1.2: Proper and Improper Rational Functions

A rational function $\frac{P(x)}{Q(x)}$ is called:

Proper if $\deg(P) < \deg(Q)$
Improper if $\deg(P) \geq \deg(Q)$

Any improper rational function can be converted to a polynomial plus a proper rational function using polynomial long division.

2. The Partial Fractions Method

Theorem 2.1: Partial Fraction Decomposition

Every proper rational function can be decomposed into a sum of simpler fractions called partial fractions.

The form of decomposition depends on the factorization of the denominator:

For each linear factor $(ax + b)$ , include $\frac{A}{ax + b}$
For each repeated linear factor $(ax + b)^n$ , include $\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \cdots + \frac{A_n}{(ax + b)^n}$
For each irreducible quadratic $ax^2 + bx + c$ , include $\frac{Ax + B}{ax^2 + bx + c}$

Standard Procedure

If improper, use polynomial division first
Factor the denominator completely
Write the partial fraction decomposition with undetermined coefficients
Clear denominators and solve for coefficients
Integrate each partial fraction separately

3. Worked Examples

Example 3.1: Simple Linear Factors

Find: $\int \frac{1}{x^2 - 1} dx$

Solution:

Step 1: Factor the denominator

x^2 - 1 = (x-1)(x+1)

Step 2: Write partial fraction decomposition

\frac{1}{x^2 - 1} = \frac{A}{x-1} + \frac{B}{x+1}

Step 3: Clear denominators

1 = A(x+1) + B(x-1)

Step 4: Solve for coefficients

Setting $x = 1$ : $1 = 2A \Rightarrow A = \frac{1}{2}$

Setting $x = -1$ : $1 = -2B \Rightarrow B = -\frac{1}{2}$

Step 5: Integrate

\int \frac{1}{x^2 - 1} dx = \int \left(\frac{1/2}{x-1} - \frac{1/2}{x+1}\right) dx

= \frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C

= \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C

Example 3.2: Repeated Linear Factors

Find: $\int \frac{2x + 1}{x^2(x-1)} dx$

Solution:

Step 1: The denominator is already factored: $x^2(x-1)$

Step 2: Write partial fraction decomposition

For the repeated factor $x^2$ , we need both $\frac{A}{x}$ and $\frac{B}{x^2}$ :

\frac{2x + 1}{x^2(x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1}

Step 3: Clear denominators

2x + 1 = Ax(x-1) + B(x-1) + Cx^2

Step 4: Solve for coefficients

Setting $x = 0$ : $1 = -B \Rightarrow B = -1$

Setting $x = 1$ : $3 = C \Rightarrow C = 3$

Comparing $x^2$ coefficients: $0 = A + C \Rightarrow A = -3$

Step 5: Integrate

\int \frac{2x + 1}{x^2(x-1)} dx = \int \left(-\frac{3}{x} - \frac{1}{x^2} + \frac{3}{x-1}\right) dx

= -3\ln|x| + \frac{1}{x} + 3\ln|x-1| + C

= 3\ln\left|\frac{x-1}{x}\right| + \frac{1}{x} + C

Example 3.3: Irreducible Quadratic Factor

Find: $\int \frac{x}{(x-1)(x^2+1)} dx$

Solution:

Step 1: The denominator has factors $(x-1)$ and $(x^2+1)$

Step 2: Write partial fraction decomposition

For the quadratic factor, we need $Ax + B$ in the numerator:

\frac{x}{(x-1)(x^2+1)} = \frac{A}{x-1} + \frac{Bx + C}{x^2+1}

Step 3: Clear denominators

x = A(x^2+1) + (Bx + C)(x-1)

Step 4: Solve for coefficients

Setting $x = 1$ : $1 = 2A \Rightarrow A = \frac{1}{2}$

Comparing $x^2$ coefficients: $0 = A + B \Rightarrow B = -\frac{1}{2}$

Setting $x = 0$ : $0 = A - C \Rightarrow C = \frac{1}{2}$

Step 5: Integrate

\int \frac{x}{(x-1)(x^2+1)} dx = \int \left(\frac{1/2}{x-1} + \frac{-x/2 + 1/2}{x^2+1}\right) dx

= \frac{1}{2}\ln|x-1| - \frac{1}{4}\ln(x^2+1) + \frac{1}{2}\arctan x + C

Example 3.4: Improper Rational Function

Find: $\int \frac{x^3 + 2x^2 + 3}{x^2 + 1} dx$

Solution:

Step 1: This is improper since $\deg(P) = 3 > \deg(Q) = 2$

Perform polynomial division:

\frac{x^3 + 2x^2 + 3}{x^2 + 1} = x + 2 + \frac{-x + 1}{x^2 + 1}

Step 2: Integrate each part

\int \frac{x^3 + 2x^2 + 3}{x^2 + 1} dx = \int \left(x + 2 + \frac{-x + 1}{x^2 + 1}\right) dx

Step 3: For the proper fraction part, split it:

\frac{-x + 1}{x^2 + 1} = \frac{-x}{x^2 + 1} + \frac{1}{x^2 + 1}

Step 4: Final result

= \frac{x^2}{2} + 2x - \frac{1}{2}\ln(x^2 + 1) + \arctan x + C

4. Special Techniques

Technique 1: Numerator Manipulation

Sometimes we can manipulate the numerator to match the derivative of the denominator:

\int \frac{2x}{x^2 + 1} dx = \ln(x^2 + 1) + C

This works because the numerator is the derivative of the denominator.

Technique 2: Completing the Square

For quadratics in the denominator, completing the square can simplify the integral:

\int \frac{1}{x^2 + 2x + 5} dx = \int \frac{1}{(x+1)^2 + 4} dx

Then use the arctangent formula.

Frequently Asked Questions

When should I use partial fractions?

Use partial fractions whenever you have a rational function (polynomial divided by polynomial) that you need to integrate. It's especially useful when the denominator can be factored.

How do I factor the denominator?

Start by looking for common factors, then try factoring as a difference of squares, sum/difference of cubes, or using the quadratic formula. Remember that some quadratics are irreducible (cannot be factored with real coefficients).

What if the partial fraction coefficients are fractions?

That's perfectly fine! Fractional coefficients are common in partial fractions. Just make sure to simplify your final answer when possible.

Practice Suggestions

• Start with simple linear factors before moving to repeated and quadratic factors
• Always check your factorization before setting up partial fractions
• Verify your answer by differentiating the result
• Practice recognizing when a function can be integrated directly without partial fractions

Practice Problems

Practice Set 1

Build confidence with rational function and substitution practice