Integration by Partial Fractions: Method and Examples

A rational function — one polynomial divided by another — is often impossible to integrate as written. But if the denominator can be factorised into linear factors, the fraction can be split into a sum of simpler fractions, each with a single linear denominator. That splitting is partial fraction decomposition, and once done, every piece integrates to a logarithm via the standard form ∫ dx/(x − a) = ln|x − a| + C.

This method matters because rational functions describe real systems everywhere: electrical circuits (Laplace transforms), chemical reaction rates, and population models all produce integrals of fractions. Partial fractions are the standard key that unlocks them, and they appear in nearly every board and competitive exam.

Key Concepts

• Proper fraction first: the method applies directly when the degree of the numerator is less than the degree of the denominator; otherwise divide the polynomials first.
• Linear factors: for denominator (x − a)(x − b), write P(x)/[(x−a)(x−b)] = A/(x−a) + B/(x−b).
• Repeated factors: (x − a)² needs A/(x−a) + B/(x−a)².
• Quadratic factors: an irreducible quadratic needs (Ax + B)/(quadratic).
• Finding constants: multiply through by the denominator, then compare coefficients or substitute the roots to solve for A, B, and so on.

Formulas

• ∫ dx/(x − a) = ln|x − a| + C
• ∫ dx/(x − a)² = −1/(x − a) + C
• ∫ (2ax + b)/(x² + bx + c) dx = ln|x² + bx + c| + C (numerator is the derivative of the denominator)

Worked Example

Evaluate ∫ (2x + 1)/(x² − x − 2) dx. Step 1: factorise the denominator: x² − x − 2 = (x − 2)(x + 1). Step 2: decompose: (2x + 1)/[(x−2)(x+1)] = A/(x−2) + B/(x+1). Step 3: multiply through: 2x + 1 = A(x + 1) + B(x − 2). Put x = 2: 5 = 3A, so A = 5/3. Put x = −1: −1 = −3B, so B = 1/3. Step 4: integrate: ∫ 5/[3(x−2)] dx + ∫ 1/[3(x+1)] dx = (5/3) ln|x − 2| + (1/3) ln|x + 1| + C. Differentiating each logarithm term recovers the original fraction, confirming the decomposition.

Common Misconceptions

• Decomposing without factorising fully: an unfactored quadratic in the denominator needs the (Ax + B) form, not a simple constant numerator.
• Skipping polynomial division: when the numerator's degree is ≥ the denominator's, divide first — partial fractions apply only to the remainder.
• Dropping the absolute value: ln|x − a| needs the modulus because a logarithm's argument must be positive.

Real-World Applications

Control engineers use partial fractions to invert Laplace transforms and analyse how circuits and mechanical systems respond over time. In chemistry, the integrated rate laws of second-order reactions produce exactly these rational integrals. Economists meet them in models of adoption curves and market saturation.

Partial fractions are divide-and-conquer for rational functions: factorise, split into simple pieces, and integrate each piece into a logarithm. Always verify your decomposition by recombining the fractions before integrating.