Integrating Functions Using Long Division and Completing the Square

Integration rules work on simple shapes: a polynomial, or a fraction where the numerator is the denominator's derivative. Messy rational functions don't start in either shape, so you reshape them. If the numerator's degree is at least the denominator's, the fraction is top-heavy, like an improper fraction such as 7/2. Long division splits it into a polynomial plus a proper remainder fraction, and each piece integrates separately: polynomial terms by the power rule, the remainder often by the log rule.

A quadratic denominator that doesn't factor nicely, like x² + 4x + 8, hides a perfect-square-plus-constant structure. Completing the square rewrites it as (x+2)² + 4, turning the integral into the arctangent pattern: the antiderivative of 1/(u²+m²) du is (1/m) arctan(u/m) + C. This works because the denominator is a sum of squares, never zero, with no derivative-matching numerator available, so a log is impossible and arctangent is the only antiderivative that fits.

Both techniques are preparation moves. They don't integrate anything by themselves; they convert an unfamiliar fraction into pieces that match rules already known: power rule, log rule, arctangent rule. This topic closes out the antidifferentiation toolkit before the course turns to accumulation and the Fundamental Theorem, where recognizing which rule an integrand matches becomes the main skill being tested in every setup.