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\[
\text{In your own words, interpret } \int \frac{du}{u} \text{ in the context and give reasons for your answer.}
\]

### Interpretation and Solution:

The expression \(\int \frac{du}{u}\) is a standard integral in calculus. This integral is known as the natural logarithm integral because its solution is the natural logarithm of \(u\). Specifically:

\[
\int \frac{du}{u} = \ln |u| + C
\]

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

### Contextual Interpretation:
- **Context**: The integral \(\int \frac{du}{u}\) typically appears in problems involving exponential growth/decay, logarithmic functions, or when using substitution in more complex integrals. 
- **Reasoning**: The reason \(\ln |u|\) is the result is because the derivative of \(\ln |u|\) is \(\frac{1}{u}\), matching the integrand \(\frac{1}{u}\).

This integral often helps in solving differential equations or simplifying expressions involving logarithmic differentiation.

Would you like further details or have any specific questions?

### Further Questions:
1. How is \(\int \frac{du}{u}\) used in solving differential equations?
2. Can you explain the substitution method in integrals using an example?
3. How does the natural logarithm function relate to exponential functions?
4. What are some practical applications of logarithms in real-life situations?
5. Could you show how this integral is used in finding the area under a curve?

### Tip:
When dealing with integrals involving logarithmic functions, always remember to include the absolute value inside the logarithm to account for both positive and negative values of \(u\).

In your own words, interpret \( \int \frac{du}{u} \) in the context and give reasons for your answer.

Explore the integral \( \int \frac{du}{u} \) and its solution as the natural logarithm of u. This calculus problem delves into logarithmic functions, their derivation, and their application in solving complex integrals. Suitable for undergraduate students studying calculus and mathematical analysis.

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