Prerequisite: Degree of numerator must be strictly less than degree of denominator. If not, perform polynomial long division first.

Type 1: Distinct Linear Factors (px+q)/((x-a)(x-b)) = A/(x-a) + B/(x-b) Find A and B by substituting x = a and x = b (cover-up method).

Type 2: Repeated Linear Factors (px+q)/(x-a)^n = A1/(x-a) + A2/(x-a)^2 + ... + An/(x-a)^n Find An by substituting x = a. Find others by comparing coefficients or successive substitution.

Type 3: Irreducible Quadratic Factor (px^2+qx+r)/((x-a)(x^2+bx+c)) = A/(x-a) + (Bx+C)/(x^2+bx+c) Find A by substituting x = a. Find B and C by comparing coefficients.

After Decomposition — Integration:

• A/(x-a) integrates to A*ln|x-a|
• A/(x-a)^n integrates to A*(x-a)^(1-n)/(1-n) for n > 1
• (Bx+C)/(x^2+bx+c): complete the square in denominator, split into ln + arctan parts

Example: integral (2x+1)/((x-1)(x^2+1)) dx Decompose: 2x+1 = A(x^2+1) + (Bx+C)(x-1) x = 1: 3 = 2A, so A = 3/2 Compare x^2 coefficients: 0 = A + B, so B = -3/2 Compare constants: 1 = A - C, so C = 1/2 integral = (3/2)ln|x-1| + integral (-3x/2 + 1/2)/(x^2+1) dx = (3/2)ln|x-1| - (3/4)ln(x^2+1) + (1/2)arctan(x) + C

Cover-up Method (Heaviside): For distinct linear factors, cover (x-a) in the original fraction and substitute x = a to find the coefficient A. This is the fastest approach for simple cases.