`int dx/ sqrt (x^2+4x+4)`

Options

A. imaginary roots
B. both the roots positive
C. roots of opposite sign
D. both the roots negative.

Explanation

To solve the integral \( \int \frac{dx}{\sqrt{x^2 + 4x + 4}} \), we can follow these steps:

### Step 1: Simplify the expression under the square root
First, we recognize that the expression \( x^2 + 4x + 4 \) is a perfect square. We can factor it as follows:

\[ x^2 + 4x + 4 = (x + 2)^2 \]

### Step 2: Rewrite the integral
Now we can rewrite the integral using this factorization:

\[ \int \frac{dx}{\sqrt{x^2 + 4x + 4}} = \int \frac{dx}{\sqrt{(x + 2)^2}} \]

### Step 3: Simplify the square root
Since the square root of a square is the absolute value, we have:

\[ \sqrt{(x + 2)^2} = |x + 2| \]

However, for the sake of integration, we can assume \( x + 2 \) is positive (or we can consider the absolute value later if needed). Thus, we simplify the integral to:

\[ \int \frac{dx}{|x + 2|} = \int \frac{dx}{x + 2} \]

### Step 4: Integrate
Now we can integrate:

\[ \int \frac{dx}{x + 2} = \ln |x + 2| + C \]

### Final Answer
Thus, the final answer to the integral is:

\[ \int \frac{dx}{\sqrt{x^2 + 4x + 4}} = \ln |x + 2| + C \]