Evaluate `int(dx)/(sqrt(x^(2)+2x+2))`.

Answer

To evaluate the integral

\[ \int \frac{dx}{\sqrt{x^2 + 2x + 2}}, \]

we can follow these steps:

### Step 1: Simplify the expression under the square root

First, we rewrite the expression \(x^2 + 2x + 2\) in a more manageable form. We can complete the square:

\[ x^2 + 2x + 2 = (x^2 + 2x + 1) + 1 = (x + 1)^2 + 1. \]

### Step 2: Substitute to simplify the integral

Now, we substitute \(t = x + 1\). Then, \(dx = dt\). The integral becomes:

\[ \int \frac{dt}{\sqrt{t^2 + 1}}. \]

### Step 3: Use the standard integral formula

We know that:

\[ \int \frac{dt}{\sqrt{t^2 + a^2}} = \ln |t + \sqrt{t^2 + a^2}| + C, \]

for \(a = 1\). Therefore, we can write:

\[ \int \frac{dt}{\sqrt{t^2 + 1}} = \ln |t + \sqrt{t^2 + 1}| + C. \]

### Step 4: Substitute back to original variable

Now we substitute back \(t = x + 1\):

\[ \ln |(x + 1) + \sqrt{(x + 1)^2 + 1}| + C. \]

### Final Result

Thus, the final answer for the integral

\[ \int \frac{dx}{\sqrt{x^2 + 2x + 2}} = \ln |x + 1 + \sqrt{(x + 1)^2 + 1}| + C. \]