91Ó°ÊÓ

A definite integral is used to calculate the area under a curve between two specified limits on the x-axis. Here, limits are represented as the numbers at the bottom and top of the integral sign. In this exercise, we are dealing with the integral from 1 to 4.

• The lower limit is 1.
• The upper limit is 4.

To solve a definite integral, follow these steps:1. **Find the Antiderivative**: This is the reverse of differentiation. For our problem, it’s the function whose derivative gives \( \frac{\ln x}{x} \).2. **Evaluate at Limits**: Substitute the upper and lower limits into the antiderivative. Subtract the result of the lower limit from the upper limit.In the step-by-step solution, we derived that:\[\int_{1}^{4} \frac{\ln x}{x} \, dx = \left[ \frac{(\ln x)^2}{2} \right]_{1}^{4}\] This formula helps compute the area under the curve from 1 to 4, or in simpler terms, the accumulated value from x=1 to x=4.

Logarithmic integration deals with functions involving natural logarithms, such as \( \ln x \). It often appears when the integrand has the form \( \frac{\ln x}{x} \), which is a standard integral form. This form is essential because it simplifies to another function we can integrate easily. Understanding logarithmic properties is crucial:

• **Natural Logarithm (ln)**: \( \ln x \) is the power to which e (approximately 2.718) must be raised to get x.
• **Integration Formula**: \( \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C \) is a key formula to solve these integrals.

In the original solution, once we isolated \( \ln x \), the integration process became straightforward using this formula. The definite integral ended up being \( 2(\ln 2)^2 \), indicating that logarithmic integration not only simplifies the process but also helps pinpoint the result accurately.

The change of base formula is a crucial tool in simplifying logarithms with different bases. If a logarithm is not in base 10 or base e, your work may require rewriting it using this formula.Consider the formula:\[\log_{b} a = \frac{\ln a}{\ln b}\]What does this formula achieve?

• **Consistent Bases**: It transforms a logarithm of any base into a natural logarithm, making calculations easier.
• **Simplification**: Aids in evaluating or integrating functions involving logs of uncommon bases by changing them into a base we can calculate easily, usually natural logarithms (base e).

In our exercise, we started with \( \log_{2} x \) and used the change of base formula:\[\log_{2} x = \frac{\ln x}{\ln 2}\]This allowed the \( \ln 2 \) terms to cancel, simplifying the integral. Such simplifications are essential for making the problem more manageable and focusing on the core integration task.