To evaluate the integral

$\int \frac{x^2 + 1}{x^2 + 2x + 2} \, dx$

we can simplify the integrand by breaking it down into more manageable parts. Here's the step-by-step process:

Step 1: Complete the square for the denominator

The denominator $x^2 + 2x + 2$ can be rewritten by completing the square:

$x^2 + 2x + 2 = (x + 1)^2 + 1$

This simplifies the expression:

$\int \frac{x^2 + 1}{(x + 1)^2 + 1} \, dx$

Step 2: Split the numerator

The numerator $x^2 + 1$ can be written as:

$x^2 + 1 = \left[ (x + 1)^2 - 1 \right] + 2$

Thus, we can rewrite the integral as:

$\int \frac{(x+1)^2 - 1 + 2}{(x + 1)^2 + 1} \, dx$

$= \int \frac{(x+1)^2 - 1}{(x+1)^2 + 1} \, dx + \int \frac{2}{(x+1)^2 + 1} \, dx$

Step 3: Solve each part separately

First part:

$\int \frac{(x+1)^2 - 1}{(x+1)^2 + 1} \, dx$

This simplifies to:

$\int 1 \, dx - \int \frac{1}{(x+1)^2 + 1} \, dx$

The first integral is straightforward:

$\int 1 \, dx = x$

The second integral is a standard arctangent integral:

$\int \frac{1}{(x+1)^2 + 1} \, dx = \tan^{-1}(x+1)$

Second part:

$\int \frac{2}{(x+1)^2 + 1} \, dx$

This is another arctangent integral:

$\int \frac{2}{(x+1)^2 + 1} \, dx = 2 \tan^{-1}(x+1)$

Step 4: Combine results

Putting everything together, the integral becomes:

$x - \tan^{-1}(x+1) + 2 \tan^{-1}(x+1) + C$

Simplifying:

$\int \frac{x^2 + 1}{x^2 + 2x + 2} \, dx = x + \tan^{-1}(x+1) + C$

Final Answer:

$\boxed{x + \tan^{-1}(x+1) + C}$

Would you like more details on any of the steps or further clarification?

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Here are 5 related questions to explore further:

1. How do you complete the square in other types of quadratic expressions?
2. What are other standard forms of integrals involving trigonometric functions?
3. How can we generalize integrals with quadratic expressions in the denominator?
4. What is the role of substitution in solving integrals like this one?
5. How do arctangent integrals arise in different calculus problems?

Tip: When working with integrals involving quadratic expressions, completing the square is a powerful technique to simplify the problem.