The given expression is:

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

We are tasked with finding the indefinite integral of this function. 

### Step-by-step Solution:

1. **Substitute** \( u = x^2 + 4 \), which implies \( du = 2x \, dx \). So, \( x \, dx = \frac{du}{2} \).

2. The integral becomes:
   \[
   \int \frac{3 \, dx}{x(x^2 + 4)^2} = \frac{3}{2} \int \frac{du}{u^2}
   \]

3. The integral of \( \frac{1}{u^2} \) is \( -\frac{1}{u} \).

4. Substituting back \( u = x^2 + 4 \), we get:
   \[
   -\frac{3}{2} \cdot \frac{1}{x^2 + 4}
   \]

5. Therefore, the indefinite integral is:
   \[
   -\frac{3}{2(x^2 + 4)} + C
   \]
   where \( C \) is the constant of integration.

Would you like more details on any specific part of this solution?

### Related Questions:

1. How would you approach a similar integral with a different denominator, say \( x^2 + a \)?
2. What changes if the power of \( (x^2 + 4) \) was different, like 3 or higher?
3. Can we solve this integral using partial fractions?
4. How can we compute the definite integral of this expression over a specific range?
5. What is the geometric interpretation of the integral \( \frac{1}{x^2 + 4} \)?

### Tip:
When solving integrals with quadratic terms, trigonometric substitution often simplifies the process, especially when the quadratic expression resembles a trigonometric identity like \( 1 + \tan^2(\theta) = \sec^2(\theta) \).

Explore the step-by-step solution for integrating the function 3/(x(x^2 + 4)^2) using the substitution method. This guide covers essential calculus concepts including substitution and integration techniques suitable for undergraduate calculus students.

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