Short on time? This page is the recap. For the full method, the graph and worked examples, use the full study guide. BC-only material. Prerequisites (FTC 6.7, ∫ 1/x dx 6.8, substitution 6.9, long division 6.10) are on the Calculus BC roadmap.
Recap
- Partial fractions is adding fractions backwards: split one hard fraction into simple fractions with linear denominators.
- Use it when the fraction is proper and the bottom factors into different linear factors.
- Try substitution first. If the top is a multiple of the derivative of the bottom, you get a single ln straight away.
- The course does not use repeated factors such as (x − 1)² or quadratic factors such as x² + 1.
Key relationships
| Step | What to write | Note |
|---|---|---|
| Decompose | N(x)/((x − p)(x − q)) = A/(x − p) + B/(x − q) | Three factors: add C/(x − r) |
| Clear fractions | N(x) = A(x − q) + B(x − p) | True for every x |
| Find constants | Put x = p, then x = q; or match coefficients | Cover-up is the quick version |
| Integrate | ∫ A/(x − p) dx = A ln|x − p| + C | Keep the absolute values |
| Non-monic factor | ∫ 1/(ax + b) dx = (1/a) ln|ax + b| + C | Chain-rule factor 1/a |
| Improper fraction | Divide first, then decompose the remainder | Needed when top degree ≥ bottom degree |
| Example | 3/(x(x + 3)) = 1/x − 1/(x + 3) | x = 0 gives A = 1; x = −3 gives B = −1 |
Mistakes to avoid
- Log of the denominator. ∫ 1/(x² − 4) dx is not ln|x² − 4|; differentiate to see why.
- Wrong root. The factor (x + 3) is zero at x = −3.
- Missing 1/a for a factor like 2x − 1.
- Decomposing before dividing when the fraction is improper.
- Limits across an asymptote. If a root of the bottom lies between the limits, the integral is improper (Topic 6.13).
- No check. Recombine the fractions; you should get the original top.
Quick self-check
- Decompose 3/(x² + 3x). (1/x − 1/(x + 3))
- Find ∫ 1/((x − 1)(x − 2)) dx. (ln|x − 2| − ln|x − 1| + C, which is ln|(x − 2)/(x − 1)| + C)
- Evaluate ∫ from 0 to 1 of 2/((x + 1)(x + 3)) dx. (ln(3/2), about 0.405, from 1/(x + 1) − 1/(x + 3))
Next: practice questions.