6.10 Long Division & Completing the Square

Improper rational functions

When the numerator's degree is ≥ the denominator's, use polynomial long division first to rewrite the integrand as a polynomial plus a proper fraction, which is easier to integrate.

Completing the square

When a quadratic denominator doesn't factor nicely, completing the square can set up an arctangent (or logarithm) form for integration.

Key terms

Long division

Use when numerator degree ≥ denominator degree.

Result of division

A polynomial plus a proper fraction.

Completing the square

Sets up arctan/log forms for unfactorable quadratics.

Where to complete

On the quadratic denominator.

Purpose

Rewrite the integrand into an integrable form.

Try an AP-style question

∫ (x² + 1)/(x + 1) dx = ?

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