Method: Complete the square in the denominator to reduce to standard forms.
ax^2 + bx + c = a[(x + b/(2a))^2 + (c/a - b^2/(4a^2))]
Case 1: Discriminant < 0 (no real roots) Integral = (1/a) * (1/k) * arctan((x+b/(2a))/k) where k^2 = c/a - b^2/(4a^2)
Case 2: Discriminant > 0 (real roots) Factor into (x-r1)(x-r2) and use partial fractions: A/(x-r1) + B/(x-r2)
Case 3: Perfect square (discriminant = 0) Integral = -1/(a(x+b/(2a))) + C
Important standard results:
- integral dx/(x^2+a^2) = (1/a)arctan(x/a) + C
- integral dx/(x^2-a^2) = (1/2a)ln|(x-a)/(x+a)| + C