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10.1 Integration of rational functions by partial fractions
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Integration of rational functions by partial fractions
What You'll Learn
Decompose rational functions into partial fractions based on denominator factors
Identify and apply the four cases: linear factors, repeated factors, irreducible quadratics, and repeated quadratics
Solve for unknown constants using substitution or equating coefficients
Integrate partial fractions using natural logarithms and inverse tangent formulas
Apply polynomial long division when the degree of the numerator exceeds the denominator
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:
This Unit Includes
7 Video lessons
Practice exercises
Skills
Partial Fractions
Rational Functions
Integration
Factoring
Long Division
Natural Logarithm
Inverse Tangent
Polynomial Decomposition
NS Curriculum Aligned