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The given integral is:

\[
\int \frac{e^{\ln x}}{x} \, dx
\]

To find the \( u \), \( dv \), \( du \), and \( v \) for integration by parts, let's first simplify the integrand.

### Simplification:
Since \( e^{\ln x} = x \) (because \( e^{\ln x} \) and \( \ln x \) are inverse functions), the integral simplifies to:

\[
\int \frac{x}{x} \, dx = \int 1 \, dx
\]

This is a straightforward integral and doesn't require integration by parts.

### Integral Solution:
\[
\int 1 \, dx = x + C
\]

Where \( C \) is the constant of integration.

### Conclusion:
Since the integral simplifies to \( \int 1 \, dx \), it doesn't require integration by parts, and thus the \( u \), \( dv \), \( du \), and \( v \) are not needed. The final answer is \( x + C \).

Would you like further details on integration by parts or have any questions? Here are some related questions for further practice:

1. What is the integral of \( \int \ln(x) \, dx \) using integration by parts?
2. How does \( e^{\ln x} = x \) work, and why do \( e \) and \( \ln \) cancel each other out?
3. What is the general formula for integration by parts?
4. How can integration by parts be applied to the integral \( \int x e^x \, dx \)?
5. What are some common mistakes to avoid when applying integration by parts?

**Tip:** When simplifying an integrand, always check if there are any functions that can cancel each other out before applying more complex integration techniques.

Explore the concept of integration by parts with a focus on understanding U, DV, DU, and V. This explanation covers a specific example where integration by parts isn't required due to simplification of the integrand. Ideal for advanced learners seeking clarity on integration techniques.

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