The integral `int_(0)^(2) (|x+2|)/(x+2)dx` is equal to

Options

A. 1
B. 3
C. 0
D. `-1`

Answer

To solve the integral \( \int_{0}^{2} \frac{|x+2|}{x+2} \, dx \), we will follow these steps:

### Step 1: Analyze the absolute value
The expression \( |x+2| \) depends on the value of \( x \). We need to determine where \( x + 2 \) is positive or negative.

- For \( x + 2 \geq 0 \) (which is true for \( x \geq -2 \)), we have \( |x + 2| = x + 2 \).
- For \( x + 2 < 0 \) (which is true for \( x < -2 \)), we have \( |x + 2| = -(x + 2) \).

Since our limits of integration are from \( 0 \) to \( 2 \), we only need to consider the case where \( x + 2 \) is positive. Thus, for \( x \in [0, 2] \), we have:

\[ |x + 2| = x + 2 \]

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

\[ \int_{0}^{2} \frac{|x+2|}{x+2} \, dx = \int_{0}^{2} \frac{x+2}{x+2} \, dx \]

### Step 3: Simplify the integrand
The expression simplifies to:

\[ \int_{0}^{2} 1 \, dx \]

### Step 4: Evaluate the integral
Now we can evaluate the integral:

\[ \int_{0}^{2} 1 \, dx = [x]_{0}^{2} = 2 - 0 = 2 \]

### Final Answer
Thus, the value of the integral is:

\[ \int_{0}^{2} \frac{|x+2|}{x+2} \, dx = 2 \]

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