Method guideCalculus I · Unit 3AIntermediate

Partial fractions: decompose before integrating

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

LaTeX article Updated July 11, 2026

\frac{p(x)}{q(x)}=\sum\text{ simpler rational 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 example

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 these ideas

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