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.