Example 1: Three Distinct Linear Factors integral $\frac{x^2+x+1}{((x-1)}$(x-2)(x-3)) dx

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

Answer: $\frac{3}{2}$ln|x-1| - 7ln|x-2| + $\frac{13}{2}$ln|x-3| + C

Example 2: Repeated Linear Factor integral $\frac{3x+2}{(x-1)}$^2(x+1) dx = $\frac{A}{x-1}$ + $\frac{B}{x-1}$^2 + $\frac{C}{x+1}$

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

Answer: $\frac{1}{4}$ln|x-1| - $\frac{5}{2}$($\frac{1}{x-1}$) + (-1/4)ln|x+1| + C = $\frac{1}{4}$ln|$\frac{x-1}{(x+1)}$| - $\frac{5}{2(x-1}$) + C

Example 3: Irreducible Quadratic integral $\frac{2x^2+3}{((x-1)}$($x^{2+x+1}$)) dx = $\frac{A}{x-1}$ + $\frac{Bx+C}{(x^2+x+1)}$

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