When Substitution Works
Before reaching for heavy machinery, check whether a simple u-substitution handles the rational function. This works when the numerator is closely related to the derivative of the denominator.
When Long Division Is Required
When the numerator degree is greater than or equal to the denominator degree, perform polynomial long division first to obtain a polynomial plus a proper fraction.
When Partial Fractions Are Required
For proper rational functions, use partial fraction decomposition to split the integrand into simpler pieces.
Integration Results by Factor Type
Linear Denominator
Repeated Linear Factor
Irreducible Quadratic Factor
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Worked Calculus Examples
Common Integration Traps
Forgetting to do long division first
Partial fractions only work on proper fractions. Always check degrees before attempting decomposition.
Missing the absolute value in logarithms
The integral of 1/(x-a) is ln|x-a| + C, not ln(x-a) + C. The absolute value matters for the domain.
Practice Integrals
Related Guides
Frequently Asked Questions
How do you know when to use partial fractions for integration?
Use partial fractions when you have a proper rational function (numerator degree less than denominator degree) whose denominator can be factored into linear and/or irreducible quadratic factors. If the fraction is improper, perform polynomial long division first.
What is the integral of 1/(x-a)?
The integral of 1/(x-a) is ln|x-a| + C. This is the most basic result from partial fraction integration and appears whenever you decompose into distinct linear factors.
When does arctangent appear in integration?
Arctangent appears when integrating fractions with irreducible quadratic denominators of the form 1/(x²+a²). After completing the square if necessary, the integral becomes (1/a)arctan(x/a) + C.
What is the difference between substitution and partial fractions?
Substitution works when the numerator is (or can be adjusted to be) the derivative of the denominator, allowing a simple u-substitution. Partial fractions is a more general technique that works for any rational function by breaking it into simpler pieces, regardless of the numerator-denominator relationship.
Can you integrate all rational functions?
Yes. Every rational function can be integrated using a combination of polynomial long division and partial fraction decomposition. The resulting integrals always involve polynomials, logarithms, and/or arctangent functions.
What are the most common mistakes when integrating rational functions?
Common mistakes include forgetting to do long division when the fraction is improper, using the wrong partial fraction template (especially for repeated or quadratic factors), sign errors when solving for coefficients, and forgetting the absolute value in logarithm results.
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