Method guideCalculus IIIntermediate12 min read

Partial fractions: decompose before integrating

Turn a proper rational function into simpler fractions whose antiderivatives are logarithmic or inverse-trigonometric.

\frac{P(x)}{Q(x)}=\sum\text{ simpler rational terms}

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Reviewed July 11, 2026

PrerequisitesStart here

First make the fraction proper by long division, then factor the denominator completely over the real numbers before choosing decomposition terms.

The denominator determines the template

Distinct linear factors receive constants over each factor. Repeated linear factors require every power through the repetition. Irreducible quadratic factors receive linear numerators.

The template is structural; missing a repeated term makes the coefficient system impossible or misleading.

Solve coefficients efficiently

After multiplying through by the common denominator, strategic substitutions can isolate coefficients for distinct linear factors. Coefficient comparison handles the remaining terms systematically.

Check the decomposition by recombining before integrating.

Recognize the final antiderivatives

Linear denominators produce logarithms. Repeated powers use the power rule after substitution. Irreducible quadratics may require completing the square and an arctangent form.

Worked exampleTwo distinct linear factors

Evaluate ∫ 1/(x² − 1) dx.

\frac1{(x-1)(x+1)}=\frac{A}{x-1}+\frac{B}{x+1}

Use one constant for each distinct linear factor.

A=\frac12,\qquad B=-\frac12

Solve by substituting x = 1 and x = −1.

\frac12\int\frac{dx}{x-1}-\frac12\int\frac{dx}{x+1}

Integrate the decomposed terms.

Result

\boxed{\frac12\ln\left|\frac{x-1}{x+1}\right|+C}

Watch for

Common mistakes

Skipping long division when the numerator degree is too large.
Forgetting intermediate powers for repeated factors.
Using a constant numerator over an irreducible quadratic.

Keep

Three takeaways

Proper fraction first, full factorization second.
The factor type determines each numerator.
Verify the algebra before integrating.