When to use: Whenever the integrand involves 1/(quadratic), 1/sqrt(quadratic), or sqrt(quadratic).

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

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

Example 1: integral dx/(x^2 + 4x + 13) = integral dx/((x+2)^2 + 9) = integral dx/((x+2)^2 + 3^2) = (1/3)arctan((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((x+2)/3) + C

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