The entire point of partial fractions is to break a rational function into simpler pieces you know how to integrate
Proper rational function is when the degree of numerator < degree of denominator
- If it is not, perform long division
- denominator goes outside the divisor
- divide leading term and write the result at the top
- multiply back the result by the divisor and write it underneath
- subtract it
- bring down next term and repeat (divide leading terms)
- We will stop once the degree of the result is less than that of the divisor / denominator
General Notes:
- Check if integrand is rational (degree of numerator is less than degree of denominator)
- If not, perform long division
- Factor the denominator completely
| Linear Factors |
x-1 |
Setting up partial fractions - Linear gets constant numerator |
A/(x-1) |
| Repeated Factors |
(x-1)^2 |
Setting up partial fractions - Linear repeated still gets constant numerator |
A/(x-1)+B/(x-1)^2 |
| Irreducible Quadratics |
(x^2+1) ex |
Setting up partial fractions - Quadratic gets linear numerator |
Bx+C/(x^2+1) |
- Write the decomposition
- Distinct linear factors are like 1/(x-1)(x+2)
- Repeated linear factors are like 1/(x-1)^3
- Becomes A/x-1 + B/(x-1)^2 + C/(x-1)^3
- We write it within 3 different terms, because if you just wrote it once, you’re only adding one power of the term
- Irreducible quadratics are like 1/(x^2+4)
- Mixed factors - combine rules
-
If the denominator is linear, the numerator is constant (A)
-
If denominator is quadratic the numerator is linear (Ax+B)
numerator degree must be one less than denominator
- Solve for constants
- Either plug in values (roots of the denominator) to find each constant OR expand and match coefficients through systems of equations
- This means grouping the expanded multiplication we just did and matching the terms to the numerator of our original expressions constants