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

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

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

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

After Decomposition — Integration:

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

Example: integral $\frac{2x+1}{((x-1)}$($x^{2+1}$)) 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 = $\frac{3}{2}$ln|x-1| + integral $\frac{-3x/2 + 1/2}{(x^2+1)}$ dx = $\frac{3}{2}$ln|x-1| - $\frac{3}{4}$ln($x^{2+1}$) + $\frac{1}{2}$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.