Question

integral de 2x-1/x^3-x^2-x+1

Answer 1

x³-x²-x+1 =

x²(x -1) -(x +1) = 0

(x-1)(x²-1) = 0

∵

x²-1 = (x+1)(x-1)

(x-1)(x²-1) = (x-1)²(x+1)

↓

$\\ integra( \frac{2x-1}{(x-1)^2(x+1)})dx = \frac{2x-1}{(x-1)^2(x+1)} =\frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+1} \\ \\ 2x-1 = A(x-1)(x+1) + B(x+1) + C(x-1) \\ \\ 2x-1 = A(x^2-1) + Bx + B +Cx -C \\ \\ 2x-1 = Ax^2-A+Bx+B+Cx-C \\ \\ 2x-1 = x^2(A) + x(B+C) + (-A+B -C) \\ \\ A = 0 \\ B+C = 2 \\ -A+B-C = -1 \\ \\ B+C= 2 \\ B -C = -1 \\ 2B =1 \\ B = 1/2 \\ 1/2+C=2 \\ C= 3/2$

$\\ integra \frac{1/2}{(x-1)^2}dx +integral \frac{3/2}{x+1} dx \\ \\ -1/2(x-1)^-^1+3/2ln|x+1|+C \\ \\ -1/2\frac{1}{x-1} +3/2Ln|x+1|+C$