Integrating Using Partial Fractions
Simple Explanation
Once a rational function is decomposed into simple fractions of the form A/(x−a), each piece integrates directly using the natural logarithm: ∫A/(x−a) dx = A·ln|x−a| + C.
Why Do We Need It?
This is the payoff of the whole partial fraction method — a rational function that looked impossible to integrate directly becomes a sum of easy logarithm integrals.
Formula
Integrating a Partial Fraction Decomposition
∫[A/(x−a) + B/(x−b)] dx = A·ln|x−a| + B·ln|x−b| + C
Once a rational function is decomposed into simple fractions of the form A/(x−a), each one integrates directly to a natural logarithm term.
- A, B
- — the constants found during the decomposition step
- C
- — the constant of integration
When to use it: As the final step after decomposing a rational function into partial fractions.
Worked Example
Integrate using a partial fraction decomposition
Find ∫1/[(x−1)(x+2)] dx, using the decomposition (1/3)/(x−1) − (1/3)/(x+2) found earlier.
Why Does This Work?
The derivative of ln|x−a| is exactly 1/(x−a) (the absolute value handles both sides of the vertical asymptote at x=a without changing the derivative), so each partial fraction term A/(x−a) is, by definition, the derivative of A·ln|x−a|.
Real-Life Example
Modeling logistic-style population growth
Certain population growth models produce a rational function of population size that needs to be integrated to find population as a function of time.
Partial fractions turn that integral into a sum of logarithm terms — exactly the technique behind deriving the closed-form logistic growth equation.
Practice
Given 1/[(x−1)(x+1)] = (1/2)/(x−1) − (1/2)/(x+1), find ∫1/[(x−1)(x+1)] dx.
HardCommon mistake
Forgetting the absolute value bars in ln|x−a| — without them, the logarithm would be undefined whenever x−a is negative, but the antiderivative must be valid on both sides of the vertical asymptote.
Quick Review
- ∫A/(x−a) dx = A·ln|x−a| + C.
- Decompose first, then integrate each simple piece separately.
- Always include the absolute value bars inside the logarithm.