https://file.aiquickdraw.com/mathbot/file/1727350796180utefakuy.jpg

The integral in the image is:

\[
\int \frac{dx}{(x^2 + 4)\sqrt{x^2 + 9}}
\]

We will solve this step by step using substitution methods.

### Step-by-Step Solution:

1. **Substitution**:
   Let \( x = 2 \tan(\theta) \). Therefore, \( dx = 2 \sec^2(\theta) d\theta \).

   Also,
   \[
   x^2 + 4 = 4(\tan^2(\theta) + 1) = 4\sec^2(\theta)
   \]
   \[
   x^2 + 9 = 4\tan^2(\theta) + 9 = 9\sec^2(\theta)
   \]

2. **Transform the Integral**:
   Substituting \( x = 2\tan(\theta) \) and the derived expressions:
   \[
   \int \frac{dx}{(x^2 + 4)\sqrt{x^2 + 9}} = \int \frac{2\sec^2(\theta) d\theta}{4\sec^2(\theta)\cdot 3\sec(\theta)}
   \]

3. **Simplify the Integral**:
   \[
   = \frac{1}{6} \int \frac{\sec^2(\theta)}{\sec^3(\theta)} d\theta = \frac{1}{6} \int \frac{d\theta}{\sec(\theta)} = \frac{1}{6} \int \cos(\theta) d\theta
   \]

4. **Integrate**:
   \[
   = \frac{1}{6} \sin(\theta) + C
   \]

5. **Back-Substitute \( \theta \)**:
   Since \( x = 2\tan(\theta) \), we have:
   \[
   \sin(\theta) = \frac{x}{\sqrt{x^2 + 4}}
   \]

   Thus, the final solution becomes:
   \[
   \frac{1}{6} \cdot \frac{x}{\sqrt{x^2 + 4}} + C
   \]

The final result is:
\[
\boxed{\frac{x}{6\sqrt{x^2 + 4}} + C}
\]

Would you like more details on any step or have further questions?

### 5 Related Questions:
1. What is the purpose of trigonometric substitution in integrals?
2. How would this integral change if the denominator had a different expression, such as \( \sqrt{x^2 + 16} \)?
3. Can this integral be solved using a different substitution method, like hyperbolic functions?
4. How can we verify the result of this integral by differentiation?
5. What is the geometric interpretation of this integral?

### Tip:
Always check for symmetry in integrals; it can help simplify certain steps during the solving process.

Learn how to solve the integral ∫ dx / ((x² + 4)√(x² + 9)) using trigonometric substitution. This problem covers key calculus concepts such as substitution methods and integration techniques, suitable for undergraduate students.

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