Problem 62 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The integral evaluates to 2.

Step 1: Analyze the Integral

We need to evaluate the integral \( \int_{1}^{4} \frac{\log_2 x}{x} \, dx \). The challenge here is the logarithm with base 2. Let's first convert the base-2 logarithm to a natural logarithm using the change of base formula: \( \log_2 x = \frac{\ln x}{\ln 2} \).

Step 2: Substitute and Simplify

Substitute \( \log_2 x = \frac{\ln x}{\ln 2} \) in the integral: \( \int_{1}^{4} \frac{\ln x}{x \ln 2} \, dx \). Since \( \ln 2 \) is a constant, we can take it outside the integral: \( \frac{1}{\ln 2} \int_{1}^{4} \frac{\ln x}{x} \, dx \).

Step 3: Recognize the Integral Formula

The integral \( \int \frac{\ln x}{x} \, dx \) matches the derivative of \((\ln x)^2\). Recall that \( \frac{d}{dx} \left( \frac{(\ln x)^2}{2} \right) = \frac{\ln x}{x} \), which means our integral can be simplified.

Step 4: Perform the Integration

Using the relationship from the previous step, integrate: \( \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C \). The definite integral becomes \( \frac{1}{\ln 2} \left[ \frac{(\ln x)^2}{2} \right]_{1}^{4} \).

Step 5: Evaluate the Definite Integral

Compute the definite integral: \( \frac{1}{\ln 2} \left( \frac{(\ln 4)^2}{2} - \frac{(\ln 1)^2}{2} \right) \). We know \( \ln 1 = 0 \) and \( \ln 4 = 2 \ln 2 \). Substitute these values to get: \( \frac{1}{\ln 2} \left( \frac{(2 \ln 2)^2}{2} - 0 \right) \).

Step 6: Simplify and Conclude

Simplify the expression: \( \frac{1}{\ln 2} \times \frac{4 (\ln 2)^2}{2} = \frac{4 (\ln 2)^2}{2\ln 2} = 2 \ln 2 \). Therefore, the value of the integral is \( 2 \).

Key Concepts

Definite Integral

In integral calculus, a definite integral is a concept used to calculate the area under a curve between two points on the x-axis, for a given function. It provides a way to sum up infinitesimally small quantities, which together represent a total accumulated value from one boundary to another. The definite integral is expressed as \[ \int_{a}^{b} f(x) \, dx \] where \(a\) and \(b\) are the lower and upper bounds of integration, and \(f(x)\) is the function being integrated.
Unlike indefinite integrals, definite integrals calculate a numeric value rather than a general function with a constant of integration. They have practical applications in various disciplines, such as finding areas, volumes, and in physics for computing things like displacement and mass.
The computation involves evaluating an anti-derivative of \(f(x)\) at the upper limit \(b\) and subtracting its evaluation at the lower limit \(a\), which is often denoted as: \[ F(b) - F(a) \] where \(F(x)\) is the antiderivative of \(f(x)\).

Change of Base Formula

The change of base formula in logarithms helps us convert logarithms of one base into another base, making them easier to solve or integrate when necessary. For functions that involve logarithms of bases other than \(e\), often converting them into natural logarithms is useful, especially when integrating.
The formula can be expressed as:

• \( \log_b x = \frac{\ln x}{\ln b} \)

This conversion formula leverages the properties of logarithms to simplify calculations, allowing us to work in a consistent base.
In the given problem, \( \log_2 x \) was converted to natural logarithms as follows:

• \( \log_2 x = \frac{\ln x}{\ln 2} \)

This change facilitated easier integration, since the natural logarithm has well-known properties and derivative formulas.

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a logarithm with the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. It is one of the most commonly used types of logarithms in calculus, as its properties make differentiation and integration straightforward.

• The derivative of \( \ln x \) is \( \frac{1}{x} \).
• The integral of \( \frac{1}{x} \) is \( \ln |x| + C \).

These properties, among others, provide a foundation for much of integral calculus involving logarithms.
In the context of the integral problem, the natural logarithm allows for simpler manipulation of the function \( \frac{\ln x}{x} \), facilitating integration and further operations due to its clear relationship with the derivative of \( \ln x \).
The connection between integrals and derivatives of \( \ln x \) simplifies the process of finding an antiderivative or applying integration techniques.

Logarithmic Integration

Integrating functions involving logarithmic expressions can often be simplified by identifying known patterns or using specific integration techniques. One such integration method involves recognizing that the derivative of a logarithmic function, such as \( (\ln x)^2 \), leads to integrals that can be calculated directly.
In our example, we integrated \( \int \frac{\ln x}{x} \, dx \) by recognizing its derivative was related to \[ \frac{d}{dx} \left( \frac{(\ln x)^2}{2} \right) = \frac{\ln x}{x} \]This allowed us to utilize the antiderivative directly:

• \( \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C \)

Such approaches are common when working with integrals involving logarithms, highlighting the interconnectedness between derivatives and integrals.
Logarithmic integration often uses basic identities and properties to transform challenging integrals into simpler forms. Evaluating these integrals with the bounds converts them into definite integrals, yielding specific numerical results.