Evaluate `int(dx)/(1+sqrt(x^(2)+2x+2))`.

Answer

To evaluate the integral

\[ I = \int \frac{dx}{1 + \sqrt{x^2 + 2x + 2}}, \]

we will follow a systematic approach.

### Step 1: Simplify the expression under the square root
First, we simplify the expression inside the square root:

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

### Step 2: Make a substitution
Next, we will use the substitution \( t = \sqrt{x^2 + 2x + 2} \). Then, we have:

\[ t^2 = x^2 + 2x + 2. \]

Differentiating both sides with respect to \( x \):

\[ 2t \frac{dt}{dx} = 2x + 2 \implies \frac{dt}{dx} = \frac{x + 1}{t} \implies dx = \frac{t}{x + 1} dt. \]

### Step 3: Express \( x \) in terms of \( t \)
From the equation \( t^2 = (x + 1)^2 + 1 \), we can express \( x + 1 \):

\[ x + 1 = \sqrt{t^2 - 1}. \]

Thus,

\[ x = \sqrt{t^2 - 1} - 1. \]

### Step 4: Substitute back into the integral
Now we substitute \( dx \) and the expression for \( x + 1 \) into the integral:

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

### Step 5: Simplify the integral
This simplifies to:

\[ I = \int \frac{t}{\sqrt{t^2 - 1}(1 + t)} dt. \]

### Step 6: Partial fractions
Next, we can break this into partial fractions. We can express:

\[ \frac{t}{\sqrt{t^2 - 1}(1 + t)} = \frac{A}{1 + t} + \frac{B}{\sqrt{t^2 - 1}}. \]

### Step 7: Solve for constants
Multiplying through by the denominator \( \sqrt{t^2 - 1}(1 + t) \) and solving for \( A \) and \( B \) will yield the constants.

### Step 8: Integrate each term
After finding \( A \) and \( B \), we can integrate each term separately.

### Step 9: Substitute back to original variable
Finally, we substitute back \( t = \sqrt{x^2 + 2x + 2} \) to express the result in terms of \( x \).

### Final Result
The final result will be in the form of:

\[ I = \text{(some expression in terms of } x\text{)} + C, \]

where \( C \) is the constant of integration.