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The integral you provided is:

\[
\int_1^2 \frac{x}{x^2 + 4x + 20} \, dx
\]

### Solution:

We can solve this integral by using substitution. Notice that the denominator can be rewritten in a more recognizable form by completing the square. Let's go through the steps:

1. **Complete the square** for the quadratic in the denominator:
   \[
   x^2 + 4x + 20 = (x + 2)^2 + 16
   \]
   
2. **Substitute**:
   Let \( u = x^2 + 4x + 20 \), and \( du = (2x + 4)dx \), which simplifies the integration process.

Now, I'll perform the integration steps to get the result.The evaluated integral is:

\[
\int_1^2 \frac{x}{x^2 + 4x + 20} \, dx = \frac{\log(32)}{2} - \frac{\log(25)}{2} + \frac{\text{atan}\left(\frac{3}{4}\right)}{2} - \frac{\pi}{8}
\]

This is the exact result, combining logarithmic and trigonometric terms.

Would you like further details on how the steps were derived? Let me know if you have any questions.

Here are some related questions to consider:

1. How do you complete the square for any quadratic expression?
2. What is the general method of substitution for solving integrals?
3. How do inverse trigonometric functions arise in integrals?
4. How do logarithms combine in expressions like this?
5. What are typical applications of definite integrals in physics or engineering?

**Tip:** When evaluating integrals, always consider simplifying the integrand with algebraic techniques like completing the square or using substitutions.

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

Learn how to evaluate the definite integral ∫[1 to 2] (x / (x^2 + 4x + 20)) dx using substitution and completing the square techniques. This solution involves logarithmic and inverse trigonometric functions. Suitable for advanced calculus students.

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