`int``x/((x+1)(x+2))`dx

Answer

To solve the integral \( \int \frac{x}{(x+1)(x+2)} \, dx \), we will use the method of partial fractions. Here are the steps to solve the integral:

### Step 1: Set Up the Integral
Let
\[ I = \int \frac{x}{(x+1)(x+2)} \, dx \]

### Step 2: Decompose into Partial Fractions
We can express the integrand as a sum of partial fractions:
\[ \frac{x}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2} \]
where \( A \) and \( B \) are constants to be determined.

### Step 3: Clear the Denominator
Multiply both sides by \( (x+1)(x+2) \):
\[ x = A(x+2) + B(x+1) \]

### Step 4: Expand and Collect Terms
Expanding the right-hand side:
\[ x = Ax + 2A + Bx + B = (A + B)x + (2A + B) \]

### Step 5: Set Up the System of Equations
By equating coefficients from both sides:
1. For \( x \): \( A + B = 1 \)
2. For the constant term: \( 2A + B = 0 \)

### Step 6: Solve the System of Equations
From the first equation, we can express \( B \) in terms of \( A \):
\[ B = 1 - A \]
Substituting into the second equation:
\[ 2A + (1 - A) = 0 \implies 2A + 1 - A = 0 \implies A + 1 = 0 \implies A = -1 \]
Now substituting \( A \) back to find \( B \):
\[ B = 1 - (-1) = 2 \]

### Step 7: Write the Partial Fraction Decomposition
Now we can write:
\[ \frac{x}{(x+1)(x+2)} = \frac{-1}{x+1} + \frac{2}{x+2} \]

### Step 8: Integrate Each Term
Now we can integrate each term separately:
\[ I = \int \left( \frac{-1}{x+1} + \frac{2}{x+2} \right) \, dx \]
This gives:
\[ I = -\int \frac{1}{x+1} \, dx + 2\int \frac{1}{x+2} \, dx \]

### Step 9: Compute the Integrals
The integrals can be computed as follows:
\[ I = -\ln|x+1| + 2\ln|x+2| + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final answer is:
\[ I = -\ln|x+1| + 2\ln|x+2| + C \]

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