Problem 102 - Thomas Calculus

Question

Evaluate the integrals. \begin{equation}\int_{1}^{4} \frac{\log _{2} x}{x} d x\end{equation}

Step-by-Step Solution

Answer

The value of the integral is \( 2 \ln 2 \).

Step 1: Identify Substitution

To solve the integral \( \int_{1}^{4} \frac{\log _{2} x}{x} \, dx \), we'll start by rewriting \( \log_{2} x \). Recall that \( \log_{2} x = \frac{\ln x}{\ln 2} \), where \( \ln x \) is the natural logarithm. So, the integral becomes: \[ \int_{1}^{4} \frac{\ln x}{x \ln 2} \, dx. \]

Step 2: Simplify the Integral

Factor out the constant \( \frac{1}{\ln 2} \): \[ \frac{1}{\ln 2} \int_{1}^{4} \frac{\ln x}{x} \, dx \] This allows us to focus on the integral \( \int_{1}^{4} \frac{\ln x}{x} \, dx \).

Step 3: Use Substitution for Integration

Make the substitution \( u = \ln x \), then \( du = \frac{1}{x} \, dx \) which matches the differential in the integral. This changes the limits of integration: when \( x = 1, u = \ln 1 = 0 \) and when \( x = 4, u = \ln 4 \). The integral becomes: \[ \frac{1}{\ln 2} \int_{0}^{\ln 4} u \, du. \]

Step 4: Evaluate the Substituted Integral

Now integrate \( u \, du \): \[ \int u \, du = \frac{u^2}{2} + C. \] Evaluating from 0 to \( \ln 4 \), we have: \[ \frac{1}{2} (\ln 4)^2. \]

Step 5: Substitute back the Limits and Evaluate

Apply the constant: \[ \frac{1}{\ln 2} \times \frac{1}{2} (\ln 4)^2 = \frac{(\ln 4)^2}{2 \ln 2}. \] Since \( \ln 4 = 2 \ln 2 \), substitute: \[ \frac{(2 \ln 2)^2}{2 \ln 2} = \frac{4 (\ln 2)^2}{2 \ln 2}. \] Simplify to get \( 2 \ln 2 \).

Step 6: Final Calculation

The calculation simplifies to \( 2 \ln 2 \). Therefore, the value of the definite integral is \( 2 \ln 2 \).

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus that allows us to calculate the area under a curve within a given interval. Unlike indefinite integrals, which contain an arbitrary constant, definite integrals evaluate a function for a specific range of values. This means they have fixed upper and lower limits. The notation of a definite integral is usually written as \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) represent the limits of integration, and \(f(x)\) is the function being integrated. The result of a definite integral is a number, which represents the net area between the curve of \(f(x)\) and the x-axis from \(x = a\) to \(x = b\).

• Definite integrals can account for areas above and below the x-axis. If parts of the area are below the x-axis, they are considered negative while above the x-axis is positive.
• They are used not only in calculating areas but also in finding volumes, work done by a force, and other accumulative measures.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are critical in calculus and have properties that simplify complex problems. A logarithm is expressed as \( \log_{b}(x) \), which means the power to which the base \(b\) must be raised to produce the number \(x\). The natural logarithm, labeled as \( \ln(x) \), is a logarithm with the base \(e\), where \(e\) is approximately 2.71828. Logarithms have rules that make integration and differentiation easier, such as:

• \( \log_{b}(xy) = \log_{b}x + \log_{b}y \)
• \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y \)
• \( \log_{b}(x^k) = k \log_{b}x \)

Logarithmic functions appear frequently in natural and physical sciences, and their integration is a valuable skill in handling functions involving \( \ln(x) \).

Integration by Substitution

Integration by substitution is an effective technique used to simplify the integration process. It is particularly useful when dealing with composite functions where direct integration isn't straightforward. By substituting part of the integral with a single variable (often \(u\)), you can transform a complex integral into a simpler form. For example, if you have an integral like \( \int f(g(x))g'(x) \ dx \), setting \(u = g(x)\) transforms the integral into \( \int f(u) \, du \).

• When using substitution, carefully change the differential \(dx\) to \(du\) based on your substitution to match the new integral.
• Change the limits of integration to reflect values in terms of \(u\), as seen when \(x = 1\), \(u = \ln 1 = 0\), and \(x = 4\), \(u = \ln 4\).

This method not only simplifies the process but is also crucial for solving equations involving logarithmic and exponential functions.

Natural Logarithm

The natural logarithm, abbreviated as \( \ln(x) \), is a specific logarithm with the base \(e\), an irrational constant approximately equal to 2.71828. The natural logarithm is prevalent in calculus because it simplifies differentiation and integration processes significantly.

• One key property of the natural logarithm is \( \frac{d}{dx} \ln x = \frac{1}{x} \), which aids in integration, making it much simpler to handle integrals involving \( \ln(x) \).
• Natural logarithms are used extensively in growth and decay problems, particularly those involving continuous compounding and exponential growth patterns.

In our example, the conversion of \( \log_{2}(x) \) to \( \ln(x) / \ln(2) \) allowed us to use properties of \( \ln(x) \) to reach a result efficiently. This highlights how the natural logarithm acts as a bridge in solving calculus problems.