91Ó°ÊÓ

The change of variables, also known as substitution, is a powerful technique that simplifies complex integrals by transforming them into easier forms. This technique involves replacing a variable with another variable, often making the integral more approachable.

In our given problem, we were working with the integral \( \int_{1}^{e^{2}} \frac{\ln x}{x} d x \). The natural logarithm function \( \ln x \) can be tricky to integrate directly, so we used substitution to make the process easier.

**How Substitution Works**
1. **Choose a new variable**: Select a substitution that simplifies the integral. In this case, setting \( u = \ln x \) was effective because it reduced the compound structure of the integrand.
2. **Convert the differential**: Calculate \( du \) in terms of \( dx \). This ensures that the entire integral is expressed in terms of \( u \). Here, \( du = \frac{1}{x} dx \).
3. **Adjust the limits of integration**: Since we changed variables, the limits of integration must be updated to reflect the new bounds. When \( x = 1 \), \( u = 0 \), and when \( x = e^{2} \), \( u = 2 \).

This transformed our integral into a simpler form, \( \int_{0}^{2} u du \), where straightforward integration rules apply.

Definite integrals calculate the area under a curve within specified limits. It is crucial to respect the boundaries given in the problem as they dictate the precise region of integration.

For the integral \( \int_{1}^{e^{2}} \frac{\ln x}{x} dx \), the definite integral helps us find the exact numeric value for the area between \( x = 1 \) and \( x = e^2 \) for the function \( \frac{\ln x}{x} \). Here are the key aspects:

- **Limits of Integration**: These are \( 1 \) and \( e^2 \) in the original integral, but when using substitution, they transformed to \( 0 \) and \( 2 \), respectively, for the variable \( u \).
- **Evaluating Integrals**: Once simplified using substitution, we evaluated the definite integral \( \int_{0}^{2} u du \) by applying the power rule.
- **Outcome**: This leads us to a concrete numerical answer \( 2 \), representing the precise area between the given boundaries.

Logarithmic integration involves integrating functions that contain logarithms, such as \( \ln x \). Direct integration can be complicated, which is why substitution often simplifies the process.

In this exercise, \( \int \frac{\ln x}{x} dx \) was made manageable by recognizing that:\[ \text{If } u = \ln x, \text{ then } \frac{\ln x}{x} dx \text{ transforms to } u du. \]

**Steps in Logarithmic Integration**
- **Substitution Choice**: By choosing \( u = \ln x \), we turned a difficult integral into one that uses the power rule easily. This highlights the effectiveness of substitution when dealing with \( \ln x \).
- **Resulting Integral**: The transformation yielded \( \int u du \), allowing us to solve it using the fundamental integration techniques, thereby avoiding the complexities of logarithms.

Overall, logarithmic integration coupled with substitution is a strategic approach that demystifies and resolves otherwise challenging integrals.