To solve the integral

$\int \frac{x^2 + 3}{(x^2 + 2x + 3)^2} \, dx,$

we'll proceed by examining potential substitutions and simplifications.

Solution Steps:

1. Complete the Square in the Denominator: Rewrite $x^2 + 2x + 3$ by completing the square. $x^2 + 2x + 3 = (x + 1)^2 + 2$ This transforms the integral into a form that might make it easier to simplify.

2. Consider a Substitution: Let $u = x^2 + 2x + 3$. Then $du = (2x + 2) \, dx = 2(x+1) \, dx$, which implies $dx = \frac{du}{2(x+1)}$.

3. Separate Terms: Rewrite the integrand in terms of $u$ and attempt to split the fraction to simplify further.

Would you like a full, step-by-step solution, or do you have any specific part you would like to delve into?

Here are 5 relative questions to expand on this topic:

1. How do we choose an appropriate substitution for integrals with quadratic terms?
2. What are alternative methods for integrating rational functions?
3. How does completing the square aid in solving integrals?
4. What role does partial fraction decomposition play in integration?
5. How can we identify when an integral might be solved with trigonometric substitution?

Tip: Completing the square is a powerful technique for simplifying expressions within integrals, especially for quadratic polynomials in the denominator.