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The problem asks for the integral of the function:

\[
\int \frac{\sec^2 x}{\sqrt{\tan^2 x + 4}} \, dx
\]

### Solution:

1. **Substitution**: Let \( u = \tan x \), so \( du = \sec^2 x \, dx \).

2. **Rewrite the integral**: Substituting into the integral, we have:
   \[
   \int \frac{\sec^2 x}{\sqrt{\tan^2 x + 4}} \, dx = \int \frac{du}{\sqrt{u^2 + 4}}
   \]

3. **Recognize the standard form**: The integral is now in the form:
   \[
   \int \frac{du}{\sqrt{u^2 + a^2}}
   \]
   where \( a = 2 \). The solution to this integral is a known standard result:
   \[
   \int \frac{du}{\sqrt{u^2 + a^2}} = \ln\left| u + \sqrt{u^2 + a^2} \right| + C
   \]

4. **Substitute back \( u = \tan x \)**:
   \[
   \ln\left| \tan x + \sqrt{\tan^2 x + 4} \right| + C
   \]

Thus, the final answer is:

\[
\int \frac{\sec^2 x}{\sqrt{\tan^2 x + 4}} \, dx = \ln\left| \tan x + \sqrt{\tan^2 x + 4} \right| + C
\]

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

Here are five questions to expand on this topic:

1. What are other standard forms of integrals involving square roots?
2. How can trigonometric identities help simplify integrals?
3. What is the importance of the substitution method in solving integrals?
4. Can you apply partial fraction decomposition to integrals involving trigonometric functions?
5. How can this integral be verified through differentiation?

**Tip:** Familiarity with standard integral forms is crucial as it allows for quicker problem-solving by recognizing and applying known results.

Explore the integral of secant squared x over the square root of tangent squared x plus 4 with a detailed step-by-step solution. This problem covers key concepts in integration, including trigonometric substitution and standard integral forms. Suitable for advanced high school and college-level students.

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