Integration of Rational Functions (Complete Square & Improper)

Integration of rational functions extends the Further integration topic in NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-04). This subtopic handles quotients of polynomials that are not ready to integrate: a denominator such as \(x^2+2x+5\) is rewritten by completing the square, and an improper fraction, whose top degree is at least the bottom degree, is first simplified by division.

Students and teachers will cover splitting a numerator to match the derivative of the denominator and rewriting an integrand into a form that can be integrated — often giving logarithmic or inverse-tangent results — in the HSC Extension 2 course.