Problem 63 - Thomas Calculus

Question

Evaluate the integrals. \(\int_{1}^{4} \frac{\ln 2 \log _{2} x}{x} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \(2(\ln 2)^2\).

Step 1: Convert the Logarithm to Natural Logarithm

First, notice that the integrand is \(\frac{\ln 2 \log_{2} x}{x}\). Recall the change of base formula for logarithms: \( \log_{b} a = \frac{\ln a}{\ln b} \). This means \( \log_{2} x = \frac{\ln x}{\ln 2} \). Substitute this into the integrand to get: \[\int_{1}^{4} \frac{\ln 2}{x} \cdot \frac{\ln x}{\ln 2} \ dx = \int_{1}^{4} \frac{\ln x}{x} \ dx.\] The \(\ln 2\) terms cancel out.

Step 2: Recognize the Integral Form

The integral \( \int \frac{\ln x}{x} \, dx \) is standard and can be directly integrated using the formula \( \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C \). Begin to apply this formula to solve the definite integral from 1 to 4.

Step 3: Apply the Definite Integral

Using the formula from Step 2, the integral becomes:\[\int_{1}^{4} \frac{\ln x}{x} \, dx = \left[ \frac{(\ln x)^2}{2} \right]_{1}^{4}.\] This requires evaluating the antiderivative at the bounds 4 and 1.

Step 4: Evaluate the Antiderivative at the Upper Limit

Substitute \(x = 4\):\[\frac{(\ln 4)^2}{2} = \frac{(2 \ln 2)^2}{2} = \frac{4(\ln 2)^2}{2} = 2(\ln 2)^2.\]

Step 5: Evaluate the Antiderivative at the Lower Limit

Substitute \(x = 1\):\[\frac{(\ln 1)^2}{2} = \frac{0^2}{2} = 0.\]

Step 6: Subtract the Evaluations

The definite integral is the difference between the values from Steps 4 and 5:\[\int_{1}^{4} \frac{\ln x}{x} \, dx = 2(\ln 2)^2 - 0 = 2(\ln 2)^2.\] Thus, the value of the integral is \(2(\ln 2)^2\).

Key Concepts

Definite Integrals

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

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.

Change of Base Formula

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.