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We are given the following integral to solve:

\[
\int \frac{x^2 + 2x - 2}{(x^2 + 6x + 10)^2} \, dx
\]

### Step-by-step solution:

1. **Simplification of the denominator:**
   The quadratic expression in the denominator can be rewritten as a perfect square:
   \[
   x^2 + 6x + 10 = (x + 3)^2 + 1
   \]
   This suggests that a substitution of the form \( u = x + 3 \) might help simplify the integral.

2. **Substitute \( u = x + 3 \):**
   Let \( u = x + 3 \), then \( du = dx \), and the quadratic expressions transform as follows:
   \[
   x^2 + 6x + 10 = u^2 + 1
   \]
   Moreover,
   \[
   x = u - 3 \quad \Rightarrow \quad x^2 + 2x - 2 = (u - 3)^2 + 2(u - 3) - 2
   \]

   Expanding and simplifying:
   \[
   (u - 3)^2 = u^2 - 6u + 9
   \]
   \[
   2(u - 3) = 2u - 6
   \]
   So:
   \[
   x^2 + 2x - 2 = u^2 - 6u + 9 + 2u - 6 - 2 = u^2 - 4u + 1
   \]

3. **Rewrite the integral in terms of \( u \):**
   Substituting into the integral:
   \[
   \int \frac{u^2 - 4u + 1}{(u^2 + 1)^2} \, du
   \]

4. **Separate the terms:**
   The integral can be broken into three parts:
   \[
   \int \frac{u^2}{(u^2 + 1)^2} \, du - 4 \int \frac{u}{(u^2 + 1)^2} \, du + \int \frac{1}{(u^2 + 1)^2} \, du
   \]

   - For the second term, we use the substitution \( v = u^2 + 1 \), which simplifies the integral.
   - The other two integrals can be evaluated using standard techniques.

   After solving these integrals, we combine the results and back-substitute \( u = x + 3 \) to get the final answer.

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Would you like the detailed computation for each integral step or an alternative approach?

Here are 5 related questions to explore:
1. How would you evaluate integrals involving higher powers of quadratics in the denominator?
2. What are common substitutions used in rational function integrals?
3. How do you handle non-perfect square denominators in integrals?
4. What is the method of partial fractions, and how does it apply to integrals?
5. How do we use trigonometric substitutions in similar integrals?

**Tip:** Substitutions like \( u = x + a \) can simplify quadratics by reducing them to simpler forms, making the integral more manageable.

∫ (x^2 + 2x - 2) / (x^2 + 6x + 10)^2 dx

Learn how to solve the integral ∫ (x^2 + 2x - 2) / (x^2 + 6x + 10)^2 dx using the substitution method. This solution breaks down the integration of a rational function by simplifying the quadratic denominator and applying substitution. Suitable for university students studying Calculus II.

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