`I= int ( dx)/( x^(2) + 4x+ 5)`.

Answer

To solve the integral \( I = \int \frac{dx}{x^2 + 4x + 5} \), we will follow these steps:

### Step 1: Complete the square for the quadratic expression
The first step is to rewrite the quadratic expression \( x^2 + 4x + 5 \) in a completed square form.

\[ x^2 + 4x + 5 = (x^2 + 4x + 4) + 1 = (x + 2)^2 + 1 \]

### Step 2: Rewrite the integral
Now we can rewrite the integral using the completed square:

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

### Step 3: Use a substitution
Let \( u = x + 2 \). Then, \( du = dx \). The integral becomes:

\[ I = \int \frac{du}{u^2 + 1} \]

### Step 4: Recognize the standard integral
The integral \( \int \frac{du}{u^2 + 1} \) is a standard integral that equals \( \tan^{-1}(u) + C \).

### Step 5: Substitute back
Now we substitute back \( u = x + 2 \):

\[ I = \tan^{-1}(x + 2) + C \]

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

\[ I = \tan^{-1}(x + 2) + C \]

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