Method: Complete the square in the denominator to reduce to standard forms.

$ax^2$ + bx + c = a[(x + $\frac{b}{2a}$)^2 + (c/a - b^$\frac{2}{4a^2}$)]

Case 1: Discriminant < 0 (no real roots) Integral = $\frac{1}{a}$ * $\frac{1}{k}$ * arctan((x+$\frac{b}{2a}$)/k) where $k^2$ = $\frac{c}{a}$ - b^$\frac{2}{4a^2}$

Case 2: Discriminant > 0 (real roots) Factor into (x-r1)(x-r2) and use partial fractions: $\frac{A}{x-r1}$ + $\frac{B}{x-r2}$

Case 3: Perfect square (discriminant = 0) Integral = -$\frac{1}{a(x+b/(2a}$)) + C

Important standard results:

• integral $\frac{dx}{x^2+a^2}$ = $\frac{1}{a}$arctan$\frac{x}{a}$ + C
• integral $\frac{dx}{x^2-a^2}$ = $\frac{1}{2a}$ln|$\frac{x-a}{(x+a)}$| + C