When to use: Integrating P(x)/Q(x) where Q(x) factors into linear and quadratic terms.

Step 1: If deg(P) >= deg(Q), divide first to get polynomial + proper fraction.

Step 2: Factor Q(x) completely over the reals.

Step 3: Write the decomposition:

• Each factor (x-a) contributes A/(x-a)
• Each factor (x-a)^k contributes A1/(x-a) + A2/(x-a)^2 + ... + Ak/(x-a)^k
• Each factor (x^2+px+q) contributes (Ax+B)/(x^2+px+q)
• Repeated quadratic factors: similar stacking

Step 4: Find coefficients by:

• Substituting roots of Q (cover-up method for distinct linear factors)
• Comparing coefficients of powers of x
• Using convenient values of x

Step 5: Integrate each term:

• A/(x-a) => A*ln|x-a|
• A/(x-a)^n => A(x-a)^(1-n)/(1-n)
• (Ax+B)/(x^2+px+q) => split into ln and arctan parts