Partial fractions
Split a fraction with a factorised denominator into simpler fractions that are easy to integrate.
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Analogy
Breaking a large order into separate small parcels that each fit through the door.
Example
Write as (1/2)[1/(x-1) - 1/(x+1)].Check at x = 2: 1/3 = (1/2)(1 - 1/3).Integrate each part.Result: (1/2) ln|(x-1)/(x+1)| + C.
In practice
Used for rational functions, which are polynomial over polynomial.
Practice
Flashcards
- How do you split 1/((x-1)(x+1))?(1/2)[1/(x-1) - 1/(x+1)].
Questions people ask
How do I find the numbers in a partial fraction split?
Write the split with unknown constants, such as A over x minus 1 plus B over x plus 1. Multiply both sides by the common denominator, then substitute values that make terms vanish. Setting x = 1 gives A directly, and x = -1 gives B.
When do I use partial fractions?
Use them when the integrand is a polynomial divided by a polynomial that factorises, and substitution does not work. If the top has degree at least that of the bottom, divide first, then split the remainder. Long division comes first, because partial fractions only works on a proper fraction.