When to use: Whenever the integrand involves $\frac{1}{quadratic}$, 1/sqrt(quadratic), or sqrt(quadratic).

Step-by-step method: Given $ax^2$ + bx + c:

1. Factor out 'a': a[$x^2$ + $\frac{b}{a}$x + c/a]
2. Complete the square: a[(x + b/2a)^2 + (c/a - $b^2$/4$a^2$)]
3. Simplify: a[(x + b/2a)^2 + \frac{4ac-b^2}{4}$$a^2]

Example 1: integral $\frac{dx}{x^2 + 4x + 13}$ = integral $\frac{dx}{(x+2}$^2 + 9) = integral $\frac{dx}{(x+2}$^2 + 3^2) = $\frac{1}{3}$arctan$\frac{(x+2}{3}$) + C

Example 2: integral dx/sqrt(5 - 4x - $x^2$) = integral dx/sqrt(-($x^2$ + 4x - 5)) = integral dx/sqrt(-($x^2$ + 4x + 4 - 9)) = integral dx/sqrt(9 - (x+2)^2) = arcsin$\frac{(x+2}{3}$) + C

Example 3: integral $\frac{3x + 2}{(x^2 + 2x + 5)}$ dx Step 1: d/dx($x^{2+2x+5}$) = 2x+2. Write 3x+2 = $\frac{3}{2}$(2x+2) + 2-3 = $\frac{3}{2}$(2x+2) - 1. Step 2: integral = $\frac{3}{2}$*integral $\frac{2x+2}{(x^2+2x+5)}$ dx - integral $\frac{dx}{x^2+2x+5}$ Step 3: First part = $\frac{3}{2}$ln|$x^{2+2x+5}$| Step 4: Second part: $x^{2+2x+5}$ = (x+1)^2+4, so integral = $\frac{1}{2}$arctan$\frac{(x+1}{2}$) Step 5: Answer = $\frac{3}{2}$ln($x^{2+2x+5}$) - $\frac{1}{2}$arctan$\frac{(x+1}{2}$) + C