6.10 Integrating Functions Using Long Division and Completing the Square

Long division before integrating

When integrating a rational function where the degree of the numerator is greater than or equal to the degree of the denominator, use polynomial long division first. This rewrites the improper fraction as a polynomial plus a proper fraction, making each piece easier to integrate separately using basic power rule and log rules.

Completing the square for quadratics

When the denominator is an irreducible quadratic (like x² + bx + c), complete the square to rewrite it as (x − h)² + k. This transforms the integral into a form matching arctangent or arcsine antiderivative patterns, since standard forms like ∫du/(u² + a²) = (1/a)arctan(u/a) + C require this structure.

Combining both techniques

Some integrals need long division first, then completing the square on the remaining fraction's denominator. Always check the degree of numerator vs. denominator first; if numerator degree ≥ denominator degree, divide. Then examine the denominator — if it's an unfactorable quadratic, complete the square to match a known inverse trig antiderivative form.

Key terms

When should you use long division before integrating?

When the numerator's degree is greater than or equal to the denominator's degree in a rational function.

Purpose of completing the square in integration

To rewrite a quadratic denominator as (x − h)² + k, matching arctangent or arcsine antiderivative forms.

∫du/(u² + a²) =

(1/a) arctan(u/a) + C

∫du/√(a² − u²) =

arcsin(u/a) + C

Result of long division on an improper rational function

A polynomial plus a proper (remainder) fraction, each integrated separately.

Try an AP-style question

∫ (x² + 1)/(x + 1) dx = ?

1. x²/2 − x + ln|x + 1| + C
2. x²/2 + ln|x + 1| + C
3. x²/2 − x + 2 ln|x + 1| + C
4. (x² + 1) ln|x + 1| + C