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

Answer

To solve the integral \( \int \frac{x}{x+2} \, dx \), we can follow these steps:

### Step 1: Rewrite the integrand
We start with the integral:
\[ I = \int \frac{x}{x+2} \, dx \]
We can manipulate the integrand by adding and subtracting 2 in the numerator:
\[ I = \int \frac{x + 2 - 2}{x + 2} \, dx \]
This simplifies to:
\[ I = \int \left( \frac{x + 2}{x + 2} - \frac{2}{x + 2} \right) \, dx \]
Which can be rewritten as:
\[ I = \int \left( 1 - \frac{2}{x + 2} \right) \, dx \]

### Step 2: Split the integral
Now we can split the integral into two separate integrals:
\[ I = \int 1 \, dx - 2 \int \frac{1}{x + 2} \, dx \]

### Step 3: Integrate each part
Now we can integrate each part:
1. The integral of 1 with respect to \( x \):
\[ \int 1 \, dx = x \]
2. The integral of \( \frac{1}{x + 2} \):
\[ \int \frac{1}{x + 2} \, dx = \log |x + 2| \]

Putting it all together, we have:
\[ I = x - 2 \log |x + 2| + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the integral \( \int \frac{x}{x+2} \, dx \) is:
\[ \int \frac{x}{x+2} \, dx = x - 2 \log |x + 2| + C \]

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