91Ó°ÊÓ

Evaluating an integral involves finding an antiderivative, which means determining a function whose derivative is the function inside the integral. In simpler terms, integration is the reverse process of differentiation. You might encounter direct integrals like \( \int x^2 \, dx \) or ones needing special methods such as substitution or integration by parts. The goal is to express the entire function in terms of antiderivatives and constants.

• Identify the integral type: Determine whether it can be evaluated directly or requires a technique like substitution.
• Choose and apply an appropriate method: Use substitutions or algebraic manipulations as needed.
• Simplify: Ensure the result is as simple as possible with constants added for indefinite integrals.

In the given exercise, we can't directly integrate \( \frac{\ln x}{x} \) without using substitution. Hence, wrapping your head around substitution method is crucial for evaluation in such cases.

Integrating an expression involving the natural logarithm is a common task in calculus. The natural logarithm, denoted as \( \ln x \), is a logarithm with base \( e \), where \( e \approx 2.71828 \). When integrating functions involving \( \ln x \), special techniques like substitution are often used.For instance, if you encounter an integral such as \( \int \frac{\ln x}{x} dx \), direct integration isn't possible due to its structure. Here are some simplified steps to handle it:- **Substitution Setup**: Let \( u = \ln x \). This choice simplifies the integral.- **Derivative of Substitution**: This gives \( du = \frac{1}{x} dx \). This simple relation is crucial.- **Integration Process**: Following substitution, the integral becomes \( \int u \, du \), which is more straightforward to integrate.The integrability of \( \ln x \) often relies on transforming its terms via suitable substitutions, making it easier to handle within the integral.

The substitution method, also known as \( u \)-substitution, is a pivotal technique in calculus for making difficult integrals more manageable. By replacing complex parts of an integral with a single variable, this method simplifies the integration process.Here's how substitution assists in integrals like \( \int \frac{\ln x}{x} dx \):- **Why Substitute?**: The substitution method is particularly useful when the integrand contains a function and its derivative, or when simplifying a complex fraction to an elementary form.- **Choosing \( u \)**: An essential step is selecting \( u \). In our problem, letting \( u = \ln x \) simplifies the integral significantly as \( du = \frac{1}{x} dx \).- **Rewriting the Integral**: Transform the original integral into \( \int u \, du \), which is a basic power integral \( \int u \, du = \frac{u^2}{2} + C \).After integrating, re-substitute \( u \) with the original expression in terms of \( x \), giving the final solution \( \frac{(\ln x)^2}{2} + C \). This substitution method streamlines complex integral calculations, making them more accessible and solvable.