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The integral provided in the image is:

\[
\int \frac{e^x + \sin x + 4\cos x + 3}{e^x + 5\sin x + 3\cos x + 6} \, dx
\]

### Step-by-step approach:

1. **Look for a substitution or simplification**: 
   One strategy for handling integrals of this type is to check if the numerator is related to the derivative of the denominator. Let's calculate the derivative of the denominator:

   \[
   f(x) = e^x + 5\sin x + 3\cos x + 6
   \]

   Taking the derivative:

   \[
   f'(x) = e^x + 5\cos x - 3\sin x
   \]

   Notice that the numerator \(e^x + \sin x + 4\cos x + 3\) resembles the derivative of the denominator \(e^x + 5\cos x - 3\sin x\), but there is a slight difference. Let's rewrite the numerator using the derivative form:

   \[
   e^x + \sin x + 4\cos x + 3 = (e^x + 5\cos x - 3\sin x) - (\cos x + 3\sin x - 3)
   \]

2. **Decompose the integral**:
   The expression can be split into two parts:

   \[
   \int \frac{f'(x)}{f(x)} \, dx - \int \frac{\cos x + 3\sin x - 3}{f(x)} \, dx
   \]

   The first integral is straightforward and gives:

   \[
   \int \frac{f'(x)}{f(x)} \, dx = \ln |f(x)| + C = \ln |e^x + 5\sin x + 3\cos x + 6| + C
   \]

3. **Second term analysis**:
   The second part, \(\int \frac{\cos x + 3\sin x - 3}{f(x)} \, dx\), does not have an elementary solution, so we leave it in its integral form. For many cases, such expressions are left as they are unless further simplification is possible, or they can be approximated using numerical methods.

### Final Answer:

\[
\ln |e^x + 5\sin x + 3\cos x + 6| - \int \frac{\cos x + 3\sin x - 3}{e^x + 5\sin x + 3\cos x + 6} \, dx + C
\]

This is the result of the integral. Would you like more details about any of the steps, or further clarifications?

### Follow-up Questions:
1. How do you calculate the derivative of more complex functions like this one?
2. Can you explain when it's appropriate to use substitution in integration?
3. What methods are available for solving integrals that don't have elementary solutions?
4. How does the natural logarithm appear in integrals like these?
5. When is numerical integration preferred over symbolic integration?

### Tip:
When the numerator of an integrand is close to the derivative of the denominator, the logarithmic form often arises. Be mindful of such opportunities for simplification.

Solve the integral (e^x + sinx + 4cosx + 3)/(e^x + 5sinx + 3cosx + 6) step by step. This calculus problem involves substitution, logarithmic integration, and the application of the derivative of the denominator. Suitable for university-level students studying advanced integration techniques.

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