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The given integral is:

\[
\int \frac{x^3 + 2x^2 + 5x - 2}{(x^2 + 2x + 2)^2} \, dx
\]

To solve this, let's proceed step by step.

### Step 1: Substitution
We observe that the denominator is a quadratic raised to a power. A common strategy is to use substitution. Let:

\[
u = x^2 + 2x + 2
\]

Now, differentiate \( u \) with respect to \( x \):

\[
du = (2x + 2) \, dx
\]
\[
\frac{du}{2} = (x + 1) \, dx
\]

This gives us the form we can substitute.

### Step 2: Split the Numerator
Now, rewrite the numerator in terms of \( (x + 1) \):

\[
x^3 + 2x^2 + 5x - 2 = (x + 1)(x^2 + 2x + 2) + 3x - 4
\]

Notice that the term \( (x + 1)(x^2 + 2x + 2) \) corresponds directly with \( u \), which simplifies part of the integral.

### Step 3: Rewrite the Integral
Now, the integral becomes:

\[
\int \frac{(x + 1) u + (3x - 4)}{u^2} \, dx
\]

Substitute \( u \) and \( du \) accordingly:

\[
\int \frac{u \, \frac{du}{2}}{u^2} + \int \frac{(3x - 4)}{u^2} \, dx
\]

This separates into two integrals.

### Step 4: Solve the Integrals
1. The first part:

\[
\frac{1}{2} \int \frac{du}{u}
\]
This is a simple logarithmic integral:

\[
\frac{1}{2} \ln |u|
\]

2. The second part requires some more work, but it simplifies to a standard integral after substituting back for \( u \).

### Step 5: Final Expression
After performing the integration and substituting \( u = x^2 + 2x + 2 \) back, you get:

\[
\frac{1}{2} \ln |x^2 + 2x + 2| + \text{(other terms)} + C
\]

The full expression will also contain a rational function part from the second term, which simplifies after substitution.

Would you like to go into more detail about the second part of the integral or the final simplification?

### Relative Questions:
1. What is the role of substitution in integral calculus?
2. How can you handle integrals with quadratic denominators?
3. What is the significance of absolute values in logarithmic integrals?
4. How do we determine the constant of integration?
5. What techniques simplify rational integrals?

### Tip:
When faced with complex fractions in integrals, always check if substitution or splitting the numerator can simplify the problem.

Evaluate the integral: \int \frac{x^3 + 2x^2 + 5x - 2}{(x^2 + 2x + 2)^2} dx

Learn how to evaluate the integral of (x^3 + 2x^2 + 5x - 2) over (x^2 + 2x + 2)^2 using substitution. This solution involves applying the substitution method and simplifying the integrand into manageable parts, including logarithmic integration. Suitable for students studying Calculus II at the university level.

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