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 = $\frac{A}{x-a}$ + $\frac{B}{x-b}$ + $\frac{C}{x-c}$ Cover-up method: A = P(a)/[Q$\frac{x}{(x-a)}$] evaluated at x = a.

Type 2: Repeated Linear Factors Q(x) = (x-a)^n P/Q = $\frac{A_1}{x-a}$ + $\frac{A_2}{x-a}$^2 + ... + $\frac{A_n}{x-a}$^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 = $\frac{A}{x-a}$ + $\frac{Bx+C}{(x^2+bx+c)}$ The quadratic term integrates to: aln|$x^{2+bx+c}$| + barctan form (after completing the square).

Integration After Decomposition:

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

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