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The integral to evaluate is:

\[
\int_{1}^{2} \frac{x}{x^2 + 4x + 20} \, dx
\]

### Solution Steps:

1. **Complete the Square**: The denominator \(x^2 + 4x + 20\) can be rewritten by completing the square:
   
   \[
   x^2 + 4x + 20 = (x + 2)^2 + 16
   \]

2. **Substitution**: Let \( u = x + 2 \), so \( du = dx \). Also, the limits of integration change: 
   - When \( x = 1 \), \( u = 3 \)
   - When \( x = 2 \), \( u = 4 \)

   Substituting these into the integral, we get:

   \[
   \int_{3}^{4} \frac{u - 2}{u^2 + 16} \, du
   \]

3. **Split the Integral**: Now, split the integral into two parts:

   \[
   \int_{3}^{4} \frac{u}{u^2 + 16} \, du - 2 \int_{3}^{4} \frac{1}{u^2 + 16} \, du
   \]

4. **First Integral**: For the first part:

   \[
   \int \frac{u}{u^2 + 16} \, du
   \]

   This can be solved by direct substitution \( v = u^2 + 16 \), yielding:

   \[
   \frac{1}{2} \ln |u^2 + 16|
   \]

   So:

   \[
   \int_{3}^{4} \frac{u}{u^2 + 16} \, du = \frac{1}{2} \left[\ln(4^2 + 16) - \ln(3^2 + 16)\right] = \frac{1}{2} \left[\ln(32) - \ln(25)\right] = \frac{1}{2} \ln \left(\frac{32}{25}\right)
   \]

5. **Second Integral**: The second part is a standard arctangent integral:

   \[
   \int \frac{1}{u^2 + 16} \, du = \frac{1}{4} \arctan\left(\frac{u}{4}\right)
   \]

   Thus:

   \[
   -2 \int_{3}^{4} \frac{1}{u^2 + 16} \, du = -\frac{1}{2} \left[\arctan\left(\frac{4}{4}\right) - \arctan\left(\frac{3}{4}\right)\right]
   \]

   This simplifies to:

   \[
   -\frac{1}{2} \left[\arctan(1) - \arctan\left(\frac{3}{4}\right)\right]
   \]

6. **Final Result**: Putting it all together:

   \[
   \frac{1}{2} \ln\left(\frac{32}{25}\right) - \frac{1}{2} \left[\frac{\pi}{4} - \arctan\left(\frac{3}{4}\right)\right]
   \]

This is the evaluated integral. Let me know if you would like a further breakdown of any steps!

### Questions to Expand Understanding:

1. How does completing the square help simplify integrals with quadratic expressions?
2. Why is substitution useful for integrals of rational functions?
3. What is the general form of the arctangent integral?
4. How can logarithmic expressions from integration be simplified using properties of logarithms?
5. How would this integral change if different limits of integration were used?

**Tip**: Always check if you can simplify the integrand before using substitution or partial fraction decomposition!

Evaluate the integral: ∫ from 1 to 2 of (x / (x^2 + 4x + 20)) dx

This integral problem evaluates ∫ (x / (x^2 + 4x + 20)) dx by completing the square and applying substitution. The problem involves using advanced calculus techniques like the arctangent integral and substitution rule. Suitable for undergraduate students or advanced high school calculus.

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