If
`int(x^(2)-x^(3))/(x^(3)-2x^(2)-x+2)dx=Alog(c|((x+1))/(x-2)^(4)|)` then A is equal to.

Answer

To solve the integral

\[ \int \frac{x^2 - x^3}{x^3 - 2x^2 - x + 2} \, dx, \]

we start by simplifying the integrand.

### Step 1: Factor the Denominator
First, we need to factor the denominator \(x^3 - 2x^2 - x + 2\). We can check for rational roots using the Rational Root Theorem. Testing \(x = 1\):

\[ 1^3 - 2(1^2) - 1 + 2 = 1 - 2 - 1 + 2 = 0. \]

So, \(x - 1\) is a factor. We can perform polynomial long division or synthetic division to factor the cubic polynomial.

After performing synthetic division of \(x^3 - 2x^2 - x + 2\) by \(x - 1\), we get:

\[ x^3 - 2x^2 - x + 2 = (x - 1)(x^2 - x - 2). \]

Next, we factor \(x^2 - x - 2\):

\[ x^2 - x - 2 = (x - 2)(x + 1). \]

Thus, the complete factorization of the denominator is:

\[ x^3 - 2x^2 - x + 2 = (x - 1)(x - 2)(x + 1). \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral:

\[ \int \frac{x^2 - x^3}{(x - 1)(x - 2)(x + 1)} \, dx. \]

### Step 3: Simplify the Numerator
We can factor out \(x^2\) from the numerator:

\[ x^2 - x^3 = -x^3 + x^2 = -x^2(x - 1). \]

Thus, we have:

\[ \int \frac{-x^2(x - 1)}{(x - 1)(x - 2)(x + 1)} \, dx = \int \frac{-x^2}{(x - 2)(x + 1)} \, dx. \]

### Step 4: Use Partial Fraction Decomposition
Next, we perform partial fraction decomposition on \(\frac{-x^2}{(x - 2)(x + 1)}\):

\[ \frac{-x^2}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1}. \]

Multiplying through by the denominator \((x - 2)(x + 1)\) gives:

\[ -x^2 = A(x + 1) + B(x - 2). \]

Expanding and collecting like terms, we can solve for \(A\) and \(B\) by substituting convenient values for \(x\).

### Step 5: Solve for Constants
Substituting \(x = 2\):

\[ -4 = A(3) + B(0) \implies A = -\frac{4}{3}. \]

Substituting \(x = -1\):

\[ -1 = A(0) + B(-3) \implies B = \frac{1}{3}. \]

### Step 6: Rewrite the Integral
Now we can rewrite the integral:

\[ \int \left(-\frac{4}{3(x - 2)} + \frac{1}{3(x + 1)}\right) \, dx. \]

### Step 7: Integrate
Integrating term by term gives:

\[ -\frac{4}{3} \ln |x - 2| + \frac{1}{3} \ln |x + 1| + C. \]

### Step 8: Combine Logarithms
Combining the logarithms:

\[ \frac{1}{3} \ln \left| \frac{(x + 1)}{(x - 2)^{4}} \right| + C. \]

### Step 9: Compare with Given Form
Given that:

\[ \int \frac{x^2 - x^3}{x^3 - 2x^2 - x + 2} \, dx = A \ln \left| \frac{(x + 1)}{(x - 2)^{4}} \right|, \]

we can see that \(A = \frac{1}{3}\).

### Final Answer
Thus, the value of \(A\) is:

\[ \boxed{\frac{1}{3}}. \]