Evaluate: `int(x^3)/((x-1)(x-2))\ dx`

Explanation

To evaluate the integral

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

we can start by simplifying the integrand using polynomial long division.

### Step 1: Polynomial Long Division
We divide \(x^3\) by \((x-1)(x-2) = x^2 - 3x + 2\).

1. Divide the leading term: \(x^3 \div x^2 = x\).
2. Multiply \(x\) by \(x^2 - 3x + 2\):
\[ x(x^2 - 3x + 2) = x^3 - 3x^2 + 2x. \]
3. Subtract this from \(x^3\):
\[ x^3 - (x^3 - 3x^2 + 2x) = 3x^2 - 2x. \]

So, we have:
\[ \frac{x^3}{(x-1)(x-2)} = x + \frac{3x^2 - 2x}{(x-1)(x-2)}. \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral as:
\[ I = \int \left( x + \frac{3x^2 - 2x}{(x-1)(x-2)} \right) \, dx. \]

### Step 3: Integrate the First Part
The first part is straightforward:
\[ \int x \, dx = \frac{x^2}{2}. \]

### Step 4: Partial Fraction Decomposition
Now we need to integrate the second part:
\[ \int \frac{3x^2 - 2x}{(x-1)(x-2)} \, dx. \]
We will use partial fractions:
\[ \frac{3x^2 - 2x}{(x-1)(x-2)} = \frac{A}{x-1} + \frac{B}{x-2}. \]
Multiplying through by \((x-1)(x-2)\) gives:
\[ 3x^2 - 2x = A(x-2) + B(x-1). \]
Expanding the right side:
\[ 3x^2 - 2x = Ax - 2A + Bx - B = (A + B)x + (-2A - B). \]

### Step 5: Set Up the System of Equations
By comparing coefficients:
1. \(A + B = 3\) (coefficient of \(x\))
2. \(-2A - B = -2\) (constant term)

### Step 6: Solve the System
From the first equation:
\[ B = 3 - A. \]
Substituting into the second equation:
\[ -2A - (3 - A) = -2 \implies -2A - 3 + A = -2 \implies -A - 3 = -2 \implies -A = 1 \implies A = -1. \]
Then substituting back:
\[ B = 3 - (-1) = 4. \]

### Step 7: Rewrite the Integral
Now we can rewrite the integral:
\[ \int \frac{3x^2 - 2x}{(x-1)(x-2)} \, dx = \int \left( \frac{-1}{x-1} + \frac{4}{x-2} \right) \, dx. \]

### Step 8: Integrate
Now we integrate:
\[ \int \frac{-1}{x-1} \, dx + \int \frac{4}{x-2} \, dx = -\ln|x-1| + 4\ln|x-2|. \]

### Step 9: Combine Results
Now we combine all parts:
\[ I = \frac{x^2}{2} - \ln|x-1| + 4\ln|x-2| + C. \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{x^3}{(x-1)(x-2)} \, dx = \frac{x^2}{2} - \ln|x-1| + 4\ln|x-2| + C. \]