The given integral expression is:

\[
\int \frac{20 \, dx}{x(x^2 + 4)^2}
\]

### Step-by-Step Solution

1. **Substitution**:
   Let’s use substitution to simplify the integral. We choose \( u = x^2 + 4 \), which simplifies the denominator. 

   - Then, \( du = 2x \, dx \), so \( \frac{du}{2} = x \, dx \).

   Thus, the integral becomes:
   \[
   \int \frac{20 \, dx}{x(x^2 + 4)^2} = 10 \int \frac{du}{u^2}
   \]

2. **Integrating**:
   We now integrate:
   \[
   10 \int \frac{1}{u^2} \, du = 10 \left( -\frac{1}{u} \right) = -\frac{10}{u}
   \]

3. **Substitute back**:
   Now substitute back \( u = x^2 + 4 \):
   \[
   -\frac{10}{x^2 + 4} + C
   \]

Thus, the integral is:

\[
\int \frac{20 \, dx}{x(x^2 + 4)^2} = -\frac{10}{x^2 + 4} + C
\]

Would you like more details or have any questions?

Here are some related questions:
1. How would the solution change if the power of \( (x^2 + 4) \) in the denominator was different?
2. What happens if we solve this integral with a different method, such as partial fractions?
3. What is the importance of substitution in solving integrals like this?
4. Can we generalize this type of integral to \( x(x^2 + a^2)^n \)?
5. How would we handle this integral if it included definite limits?

**Tip:** Substitution often works best when simplifying complex functions inside the integral, especially when their derivatives are present elsewhere in the expression.

This solution provides a step-by-step approach to solve the integral ∫ 20dx/(x(x^2 + 4)^2) using the substitution method. Learn how substitution simplifies the integral into a solvable form. Suitable for students in Undergraduate Calculus I.

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