`int(dx)/(x^(2) +2x + 2) ` is equal to

Options

A. `sin^(-1) x +1 + c `
B. `sin h^(-1) x + 1 + c `
C. `tanh^(-1) x + 1 + c `
D. `tan^(-1) x + 1 + c `

Answer

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

### Step 1: Rewrite the Quadratic Expression
First, we need to rewrite the quadratic expression in the denominator \( x^2 + 2x + 2 \):
\[ x^2 + 2x + 2 = (x^2 + 2x + 1) + 1 = (x + 1)^2 + 1 \]

### Step 2: Substitute Variables
Next, we can use a substitution to simplify the integral. Let:
\[ t = x + 1 \quad \Rightarrow \quad dx = dt \]
Now, we can rewrite the integral in terms of \( t \):
\[ \int \frac{dx}{(x + 1)^2 + 1} = \int \frac{dt}{t^2 + 1} \]

### Step 3: Integrate
The integral \( \int \frac{dt}{t^2 + 1} \) is a standard integral that evaluates to:
\[ \int \frac{dt}{t^2 + 1} = \tan^{-1}(t) + C \]

### Step 4: Substitute Back
Now, we substitute back \( t = x + 1 \):
\[ \tan^{-1}(t) = \tan^{-1}(x + 1) \]

### Final Result
Thus, the final result of the integral is:
\[ \int \frac{dx}{x^2 + 2x + 2} = \tan^{-1}(x + 1) + C \]