Example 1: Three Distinct Linear Factors integral (x^2+x+1)/((x-1)(x-2)(x-3)) dx

First check: degree 2 = degree 3 - 1. OK, no long division needed. A/(x-1) + B/(x-2) + C/(x-3): x = 1: (1+1+1)/((1-2)(1-3)) = 3/((-1)(-2)) = 3/2 = A x = 2: (4+2+1)/((2-1)(2-3)) = 7/(1)(-1) = -7 = B x = 3: (9+3+1)/((3-1)(3-2)) = 13/(2)(1) = 13/2 = C

Answer: (3/2)ln|x-1| - 7ln|x-2| + (13/2)ln|x-3| + C

Example 2: Repeated Linear Factor integral (3x+2)/(x-1)^2(x+1) dx = A/(x-1) + B/(x-1)^2 + C/(x+1)

x = 1: 5/(1)(2) = 5/2 = B x = -1: -1/(-2)^2 = -1/4 = C Compare x^2 coefficients: 0 = A + C, so A = 1/4

Answer: (1/4)ln|x-1| - (5/2)(1/(x-1)) + (-1/4)ln|x+1| + C = (1/4)ln|(x-1)/(x+1)| - 5/(2(x-1)) + C

Example 3: Irreducible Quadratic integral (2x^2+3)/((x-1)(x^2+x+1)) dx = A/(x-1) + (Bx+C)/(x^2+x+1)

x = 1: 5/3 = A Compare x^2: 2 = A + B, so B = 2 - 5/3 = 1/3 Compare constants: 3 = -A + C, so C = 3 + 5/3 = 14/3

integral = (5/3)ln|x-1| + integral (x/3 + 14/3)/(x^2+x+1) dx For the second part: split (x/3 + 14/3)/(x^2+x+1): Write x/3 + 14/3 = (1/6)(2x+1) + (14/3 - 1/6) = (1/6)(2x+1) + 9/2 = (1/6)ln|x^2+x+1| + (9/2)integral dx/((x+1/2)^2 + 3/4) = (1/6)ln|x^2+x+1| + (9/2)(2/sqrt(3))arctan((2x+1)/sqrt(3)) = (1/6)ln|x^2+x+1| + 3*sqrt(3)*arctan((2x+1)/sqrt(3))