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The given integral is:

\[
\int_0^4 \frac{x}{x + 8} \, dx
\]

### Solution:

1. **Substitute:** To simplify the integral, let's use the substitution:
   \[
   u = x + 8 \quad \Rightarrow \quad du = dx
   \]
   When \(x = 0\), \(u = 8\), and when \(x = 4\), \(u = 12\).

2. **Rewrite the integral:**
   After substitution, the integral becomes:
   \[
   \int_8^{12} \frac{u - 8}{u} \, du = \int_8^{12} \left(1 - \frac{8}{u}\right) du
   \]
   This separates into two simpler integrals:
   \[
   \int_8^{12} 1 \, du - 8 \int_8^{12} \frac{1}{u} \, du
   \]

3. **Evaluate the integrals:**
   - The first integral:
     \[
     \int_8^{12} 1 \, du = u \Big|_8^{12} = 12 - 8 = 4
     \]
   - The second integral:
     \[
     \int_8^{12} \frac{1}{u} \, du = \ln|u| \Big|_8^{12} = \ln(12) - \ln(8) = \ln\left(\frac{12}{8}\right) = \ln\left(\frac{3}{2}\right)
     \]

4. **Final expression:**
   Now, subtract the two results:
   \[
   4 - 8 \ln\left(\frac{3}{2}\right)
   \]

Thus, the value of the integral is:

\[
4 - 8 \ln\left(\frac{3}{2}\right)
\]

---

Would you like a more detailed explanation or have any further questions?

### Related Questions:
1. How can we handle integrals with more complex substitutions?
2. What is the importance of using logarithms in integration?
3. How does substitution simplify complex integrals?
4. Can integration by parts be applied to solve this integral?
5. What are some common mistakes when performing substitution in integrals?

### Tip:
Always remember to update the limits of integration after a substitution to avoid converting back to the original variable.

Evaluate the integral ∫ from 0 to 4 of (x / (x + 8)) dx.

This solution demonstrates how to evaluate the integral ∫ (x / (x + 8)) dx from 0 to 4 using substitution and logarithmic integration. Step-by-step calculations show how to solve for 4 - 8 ln(3/2). Ideal for students in college-level calculus.

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