Prerequisite: Ensure deg(P) < deg(Q). If not, perform polynomial long division first.

Type 1: Distinct Linear Factors Q(x) = (x-a)(x-b)(x-c)... P/Q = A/(x-a) + B/(x-b) + C/(x-c) Cover-up method: A = P(a)/[Q(x)/(x-a)] evaluated at x = a.

Type 2: Repeated Linear Factors Q(x) = (x-a)^n P/Q = A_1/(x-a) + A_2/(x-a)^2 + ... + A_n/(x-a)^n Find A_n by cover-up at x = a. Find others by comparing coefficients or differentiating.

Type 3: Irreducible Quadratic Factor Q(x) = (x-a)(x^2+bx+c) where x^2+bx+c has no real roots P/Q = A/(x-a) + (Bx+C)/(x^2+bx+c) The quadratic term integrates to: aln|x^2+bx+c| + barctan form (after completing the square).

Integration After Decomposition:

• A/(x-a) integrates to A*ln|x-a|
• A/(x-a)^n integrates to A/[(1-n)(x-a)^(n-1)] for n > 1
• (Bx+C)/(x^2+px+q): split as (B/2)*(2x+p)/(x^2+px+q) + (C-Bp/2)/(x^2+px+q), giving logarithm + arctan terms

JEE Shortcut for (px+q)/((x-a)(x-b)): Write px+q = L(x-a) + M(x-b). Then L = (pb+q)/(b-a), M = (pa+q)/(a-b). Integral = Lln|x-a| + Mln|x-b| + C.