Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-12, evaluate the given integral. $$ \int_{1}^{2} \frac{\ln x}{x} d x $$

Step-by-Step Solution

Answer

The value of the integral is approximately 0.2402.

Step 1: Recognize the Integral Type

The integral \( \int_{a}^{b} \frac{\ln x}{x} dx \) is a standard form that can be solved using integration by parts or by recognizing it as the derivative of a function.

Step 2: Identify Suitable Method

Note that the integrand \( \frac{\ln x}{x} \) suggests the use of a substitution that simplifies the integration. Recognize that the derivative of \( \ln x \) is \( \frac{1}{x} \). Using this, we can directly identify that the integration of \( \frac{\ln x}{x} \) leads to another known function.

Step 3: Use Known Result for Interal

Recall that the integral of \( \frac{\ln x}{x} \) from 1 to \( e \) is the definition of the natural logarithm property: \( \int_{1}^{x} \frac{\ln t}{t} dt = \frac{1}{2} (\ln x)^2 \). Apply this result to directly write the solution from 1 to 2.

Step 4: Evaluate the Integral Bounds

Now compute \( \left[ \frac{1}{2} (\ln x)^2 \right]_{1}^{2} \). First, find \( \frac{1}{2} (\ln 2)^2 \) and then subtract \( \frac{1}{2} (\ln 1)^2 \).

Step 5: Simplify Expression

The natural logarithm of 1 is zero: \( \ln 1 = 0 \). So \( \frac{1}{2} (\ln 1)^2 = 0 \). Calculate \( \frac{1}{2} (\ln 2)^2 = \frac{1}{2} \times (0.6931)^2 \approx 0.2402 \).

Step 6: Final Result

Subtracting the bounds results: \( \frac{1}{2} (\ln 2)^2 - 0 = \frac{1}{2} (\ln 2)^2 \approx 0.2402 \). This is the value of the given integral.

Key Concepts

Natural Logarithm

When you hear the term natural logarithm, it's referring to logs with the base of the number e (approximately 2.718). It's often denoted as \( \ln x \). Natural logs are used across various mathematical fields because they transform multiplication operations into addition, making complex expressions easier to manage.

• Properties: The natural log of 1, \( \ln 1 \), is 0 because any number raised to the zero power is 1. For example, \( e^0 = 1 \).
• Derivative: The derivative of \( \ln x \) is \( \frac{1}{x} \). This is crucial in integration, especially when dealing with fractions like \( \frac{\ln x}{x} \). Recognizing this relationship helps identify the integration method to use.

Understanding \( \ln x \) is essential because it frequently appears in calculus, especially with continuous growth scenarios. It forms the foundation when dealing with the integration of functions involving logs. Remember that the natural log helps in simplifying problems by converting multiplication sequences into addition, easing calculations.

Integral Bounds

Integral bounds are the values that define the start and end points of integration in definite integrals. For example, in the integral \( \int_{1}^{2} \frac{\ln x}{x} dx \), 1 and 2 are the bounds. These bounds tell us where to begin our integration and where to stop.

• Importance: Computing definite integrals requires evaluating the function at these bounds. It's like calculating the area under a curve, and these bounds are the limits of that area.
• Usage: To solve, you first integrate the function as if it were indefinite, then substitute the bounds into the resulting expression. The values obtained are then subtracted.

For this problem, after finding the antiderivative \( \frac{1}{2} (\ln x)^2 \), evaluating it at 2 and 1 gives the final result. Calculating the differences of these results gives us the net value covering the specified interval.

Substitution Method

The substitution method in integration is akin to reversing the process of the chain rule in differentiation. It's a technique where you substitute part of the integrand with a new variable to simplify the integral.

• Steps: Choose a substitution that makes the integral easier. In this exercise, recognizing \( \frac{\ln x}{x} \) helps identify substitution possibilities because the derivative of \( \ln x \) is \( \frac{1}{x} \).
• Benefits: Simplification helps in reducing the complexity of the integral. Essentially, you're transforming the problem into one you can easily solve, like switching to a different tool in your toolbox.

In some integrals, the substitution directly leads to known results, as with this exercise. Recognizing patterns and standard forms of integrals aids efficient use of substitution. With practice, spotting these cues becomes faster, allowing accurate application of integration techniques.