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The question is asking to calculate the integral:

\[
\int \left( e^{\ln(x)} + \frac{1}{x} + \cos\left( \sin\left( \frac{\pi}{6} \right) x \right) \right) dx
\]

Let's break down the integral and solve it step-by-step:

1. **Simplify \( e^{\ln(x)} \):**
   \[
   e^{\ln(x)} = x
   \]

   So, the integral becomes:
   \[
   \int \left( x + \frac{1}{x} + \cos\left( \sin\left( \frac{\pi}{6} \right) x \right) \right) dx
   \]

2. **Handle each term separately:**

   - For \( \int x \, dx \):
     \[
     \int x \, dx = \frac{x^2}{2}
     \]

   - For \( \int \frac{1}{x} \, dx \):
     \[
     \int \frac{1}{x} \, dx = \ln(x)
     \]

   - For \( \int \cos\left( \sin\left( \frac{\pi}{6} \right) x \right) dx \):
     - Note that \( \sin\left( \frac{\pi}{6} \right) = \frac{1}{2} \).
     - Thus, we have \( \int \cos\left( \frac{1}{2} x \right) dx \).
     - The antiderivative of \( \cos\left( \frac{1}{2} x \right) \) is \( \frac{2}{\sin\left( \frac{\pi}{6} \right)} \sin\left( \frac{\pi}{6} x \right) \), considering the adjustment for the coefficient.

3. **Combine the results:**
   
   The integrated expression is:
   \[
   \frac{x^2}{2} + \ln(x) + \frac{\sin\left( \frac{\pi}{6} x \right)}{\sin\left( \frac{\pi}{6} \right)} + C
   \]

4. **Match with the options:**

   - This corresponds to option **(b)**:
     \[
     \frac{x^2}{2} + \ln(x) + \frac{\sin\left( \frac{\pi}{6} x \right)}{\sin\left( \frac{\pi}{6} \right)} + C
     \]

Therefore, option **(b)** is correct.

Would you like a deeper breakdown of any step, or do you have further questions?

### Related Questions
1. How would you approach an integral involving \( e^{f(x)} \) instead of \( e^{\ln(x)} \)?
2. Why does \( e^{\ln(x)} \) simplify to \( x \)?
3. What is the antiderivative of \( \cos(kx) \) for any constant \( k \)?
4. How do trigonometric substitutions simplify integrals involving sine and cosine terms?
5. How does the constant \( C \) appear in indefinite integrals?

### Tip
When simplifying integrals, always check if functions like \( e^{\ln(x)} \) or similar expressions can be simplified directly, as it can save significant calculation steps.

Calculate \( \int \left( e^{\ln(x)} + \frac{1}{x} + \cos\left( \sin\left( \frac{\pi}{6} \right) x \right) \right) dx \).

Learn how to calculate the integral \( \int \left( e^{\ln(x)} + \frac{1}{x} + \cos\left( \sin\left( \frac{\pi}{6} \right) x \right) \right) dx \) with a step-by-step solution. This problem explores integration of logarithmic and trigonometric functions, suitable for undergraduate calculus students.

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