Key Concepts and Formulas

• Integration by Parts: $\int u \, dv = uv - \int v \, du$
• Derivative of $\ln x$: $\frac{d}{dx}(\ln x) = \frac{1}{x}$

Step-by-Step Solution

Step 1: Define the integral and apply integration by parts. We are given the integral $I = \int \cos(\ln x) \, dx$. We will use integration by parts. Let $u = \cos(\ln x)$ and $dv = dx$. Then $du = -\sin(\ln x) \cdot \frac{1}{x} \, dx$ and $v = x$. Applying integration by parts: $I = \int \cos(\ln x) \, dx = x \cos(\ln x) - \int x \left( -\sin(\ln x) \cdot \frac{1}{x} \right) \, dx = x \cos(\ln x) + \int \sin(\ln x) \, dx$

Step 2: Apply integration by parts again. Now, we need to evaluate $\int \sin(\ln x) \, dx$. Let $u = \sin(\ln x)$ and $dv = dx$. Then $du = \cos(\ln x) \cdot \frac{1}{x} \, dx$ and $v = x$. Applying integration by parts: $\int \sin(\ln x) \, dx = x \sin(\ln x) - \int x \left( \cos(\ln x) \cdot \frac{1}{x} \right) \, dx = x \sin(\ln x) - \int \cos(\ln x) \, dx$

Step 3: Substitute back into the original integral. Substituting this result back into the expression for $I$ from Step 1: $I = x \cos(\ln x) + \left( x \sin(\ln x) - \int \cos(\ln x) \, dx \right) = x \cos(\ln x) + x \sin(\ln x) - \int \cos(\ln x) \, dx$ Notice that $\int \cos(\ln x) \, dx$ is just $I$. So we have: $I = x \cos(\ln x) + x \sin(\ln x) - I$

Step 4: Solve for I. Adding $I$ to both sides gives: $2I = x \cos(\ln x) + x \sin(\ln x)$ Dividing by 2: $I = \frac{x}{2} \left( \cos(\ln x) + \sin(\ln x) \right) + C$ where $C$ is the constant of integration.

Step 5: Rearrange to match the available options. We can rewrite the solution as: $I = \frac{x}{2} \left( \sin(\ln x) + \cos(\ln x) \right) + C$

Common Mistakes & Tips

• Sign Errors: Be extremely careful with the signs when applying integration by parts, especially when differentiating $\cos(\ln x)$ and $\sin(\ln x)$.
• Choosing u and dv: The choice of u and dv in integration by parts can significantly impact the complexity of the integral. In this case, choosing $dv = dx$ simplifies the subsequent integration.
• Constant of Integration: Don't forget to add the constant of integration, C, at the end of the process.

Summary

We evaluated the integral $\int \cos(\ln x) \, dx$ using integration by parts twice. By recognizing that the original integral reappeared after the second application of integration by parts, we were able to solve for the integral algebraically. This yielded the solution $I = \frac{x}{2} \left( \sin(\ln x) + \cos(\ln x) \right) + C$.

The final answer is \boxed{\frac{x}{2}[\sin(\log e x) + \cos(\log e x)] + C}, which corresponds to option (C).