Mathfolis

Partial Fractions

Unit 6 · Integration & Accumulation

Partial Fractions (BC Only)

Partial fraction decomposition breaks a rational function into simpler fractions, each of which integrates to a logarithm. The denominator must be factored into distinct linear factors first. If the numerator degree ≥ denominator degree, perform long division first.

Decomposition Form (Distinct Linear Factors)

P(x)(x−a)(x−b)=Ax−a+Bx−b\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

Strategy

1. Factor the denominator
2. Set up decomposition and clear denominators
3. Plug in strategic x values to find A, B, …
4. Integrate each term using ∫ A/(x−a) dx = A ln|x−a| + C
AP Tip: To find A or B quickly, substitute the root of each factor (e.g., x = a to eliminate B).
Type 1

Distinct Linear Factors — Full Integral

Factor the denominator, set up partial fractions, solve for the constants, then integrate.

Example 1

Evaluate the integral.

∫5x−2(x+1)(x−3) dx\int \frac{5x-2}{(x+1)(x-3)}\,dx
Example 2

Evaluate the integral.

∫4x2−4 dx\int \frac{4}{x^2-4}\,dx

Practice more of this type— AI-generated · infinite problems

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Type 2

Setting Up the Decomposition

Factor the denominator and write the partial fraction form. Solve for the constants without necessarily integrating.

Example 3

Write the partial fraction decomposition of 2x+7 over x²+x−6 and find A and B.

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