将 $\frac{2x-1}{x^2(x+1)}$ 化为部分分式。

1. State the correct form for the partial fractions

Let $\frac{2x-1}{x^2(x+1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}$

2. Form the basic algebraic identity

Multiplying throughout by $x^2(x+1)$ gives:
$2x - 1 = Ax(x+1) + B(x+1) + Cx^2$

3. Find the values of the constants by substitution

Substitute $x = 0$:
$2(0) - 1 = B(0 + 1)$
$-1 = B$

Substitute $x = -1$:
$2(-1) - 1 = C(-1)^2$
$-3 = C$

Compare coefficients of $x^2$:
$0 = A + C$
$A = -C = 3$

4. Write the final partial fraction expression

$\frac{2x-1}{x^2(x+1)} = \frac{3}{x} - \frac{1}{x^2} - \frac{3}{x+1}$