Integrating Algebraic Fractions

Simplify the fraction before choosing a method. Polynomial division produces a polynomial plus a proper rational fraction. **Denominator or numerator structure** | **Method** Numerator degree at least denominator degree | Divide first, then inspect the remainder. At most two distinct linear denominator factors | Decompose the proper fraction, then integrate logarithmic terms. A positive quadratic denominator | Complete the square and scale to an inverse-tangent form. A square root of a positive difference of squares | Complete the square and scale to an inverse-sine form. A linear numerator over a quadratic | Express the numerator as a multiple of the denominator derivative plus a constant. Partial fractions in AA HL AHL1.11 specifies a proper fraction with at most two distinct linear denominator factors. AHL5.15 applies that decomposition to integration. 3x+5(x+1)(x+2)= 2x+1+ 1x+2 3x+5(x+1)(x+2),dx=2ln|x+1|+ln|x+2|+C Keep absolute values in logarithms unless the domain establishes positivity. A definite integral cannot be evaluated across a denominator zero by simply substituting endpoints.