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Envontrar los limites de integración de esta funcion

In the image, we see a graph resembling a region below the curve \( y = \ln(x) \), where the x-axis appears to be labeled with values 0.5 and 1, and the y-axis seems to have some numerical annotations, possibly involving the natural logarithm.

To find the limits of integration for this region, we typically:

1. Identify the x-values that define the horizontal boundaries of the region (which in this case seem to be \( x = 0.5 \) and \( x = 1 \)).
2. Define the vertical limits based on the function and the x-axis (here, the curve \( y = \ln(x) \)).

### Limits of Integration:
- **Lower limit for x**: \( x = 0.5 \)
- **Upper limit for x**: \( x = 1 \)
- **Function of y**: \( y = \ln(x) \), which describes the curve.

The region you're integrating is likely from \( x = 0.5 \) to \( x = 1 \) under the curve \( \ln(x) \).

Would you like a more detailed explanation or assistance in solving the integral of this function over these limits?

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**Further questions:**

1. How do you calculate the area under the curve \( \ln(x) \)?
2. What is the significance of the points \( x = 0.5 \) and \( x = 1 \) in terms of the function \( \ln(x) \)?
3. How do you interpret the value of the natural logarithm at specific points (e.g., \( \ln(0.5) \), \( \ln(1) \))?
4. Can you explain the behavior of the logarithmic function as \( x \to 0 \) or \( x \to \infty \)?
5. How can you verify the convergence of an integral involving \( \ln(x) \)?

**Tip**: When dealing with logarithmic integrals, pay close attention to whether the function becomes undefined or diverges at the boundaries.

Discover how to find the limits of integration for the logarithmic function ln(x) in the region between x = 0.5 and x = 1. This tutorial covers key calculus concepts, including definite integrals and the natural logarithmic function. Suitable for undergraduate students.

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