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Integration of rational functions by partial fractions

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Mathematics at Work 11
Number
10. Demonstrate an understanding of credit options including credit cards and loans
10.1 Integration of rational functions by partial fractions

Integration of Rational Functions by Partial Fractions

Learn to integrate rational functions by decomposing them into partial fractions, with rules for linear, repeated, and irreducible quadratic factors.

What You'll Learn

Recognize when a rational function's degree pattern calls for partial fraction decomposition before integrating
Factor the denominator completely into linear and irreducible quadratic factors
Set up the correct partial fraction form for distinct, repeated, and quadratic factors
Solve for unknown coefficients using substitution or equating coefficients
Integrate each simpler fraction using logarithm and arctangent formulas
Check the original integrand's degree and apply polynomial division first if needed

What You'll Practice

1

Factoring denominators into linear and irreducible quadratic factors

2

Setting up partial fraction decompositions for each case

3

Solving systems of equations to find numerator constants

4

Integrating rational functions with repeated linear factors

5

Using long division to convert improper fractions before decomposition

Why This Matters

Partial fraction integration is essential for solving complex rational integrals in calculus, differential equations, and engineering applications. This technique simplifies seemingly impossible integrals into manageable pieces, and is foundational for Laplace transforms and system analysis in advanced mathematics and physics.

Before You Start — Make Sure You Can:

Partial fraction decompositionFactoring difference of squares: x^2 - y^2Factoring trinomialsPolynomial synthetic division Completing the square

This Unit Includes

7 Video lessons
Practice exercises
Learning resources

Skills

Partial Fractions
Rational Functions
Integration
Factoring
Long Division
Natural Logarithm
Inverse Tangent
Polynomial Decomposition
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