Problem 36 - University Calculus: Early Transcendentals

Question

Guess an antiderivative for the integrand function. Validate your guess by differentiation and then evaluate the given definite integral. (Hint: Keep in mind the Chain Rule in guessing an antiderivative. You will learn how to find such antiderivatives in the next section.) $$\int_{1}^{2} \frac{\ln x}{x} d x$$

Step-by-Step Solution

Answer

The value is \(\frac{(\ln 2)^2}{2}\).

Step 1: Identify the Integrand

The given integrand is \( \frac{\ln x}{x} \). We need to find an antiderivative for this function.

Step 2: Guess the Antiderivative

Notice that the derivative of \( \ln x \) is \( \frac{1}{x} \). By considering the structure of the integrand, we can guess that an antiderivative of \( \frac{\ln x}{x} \) might be \( (\ln x)^2/2 \).

Step 3: Validate the Antiderivative

Differentiate \( (\ln x)^2/2 \) with respect to \( x \):\[ \frac{d}{dx} \left( \frac{(\ln x)^2}{2} \right) = \frac{1}{2} \cdot 2 \ln x \cdot \frac{1}{x} = \frac{\ln x}{x}. \]This confirms our guess is correct, as the derivative matches the integrand.

Step 4: Evaluate the Definite Integral

Now, evaluate the definite integral:\[ \int_{1}^{2} \frac{\ln x}{x} \, dx = \left[ \frac{(\ln x)^2}{2} \right]_1^2. \]Calculate:1. For \( x = 2 \), \( \frac{(\ln 2)^2}{2} \).2. For \( x = 1 \), \( \frac{(\ln 1)^2}{2} = 0 \) because \( \ln 1 = 0 \).Subtract:\[ \frac{(\ln 2)^2}{2} - 0 = \frac{(\ln 2)^2}{2}. \]

Step 5: Final Answer

The value of the definite integral from 1 to 2 of \( \frac{\ln x}{x} \) is \( \frac{(\ln 2)^2}{2} \). This is the evaluated result.

Key Concepts

Antiderivatives

Finding antiderivatives is like uncovering the original function from its rate of change. In our problem, the core task is to determine the antiderivative of the function \( \frac{\ln x}{x} \).
This process is essentially reversing differentiation. How do we approach this?

• First, understand that the antiderivative of a function is a new function whose derivative is the original function.
• For example, differentiating \( \ln x \) gives \( \frac{1}{x} \), so reversing it gives insight into the antiderivative of related functions.

In our exercise, we guessed the antiderivative as \( \frac{(\ln x)^2}{2} \). The guessing here involves recognizing the structure of the original function and intelligently selecting a function that differentiates back to it.

Chain Rule

The chain rule is a vital tool in both differentiation and integration. It helps to differentiate compositions of functions, essentially functions within functions.
Why is it important?

• Because it allows us to handle complex expressions like \( (\ln x)^2 \)x within differentiation.
• In our exercise, differentiating \( \frac{(\ln x)^2}{2} \) requires the chain rule.

Here's how it was applied: when differentiating \( (\ln x)^2 \), we multiply the derivative of \( \ln x \) (which is \( \frac{1}{x} \)) by the derivative of \( x^2 \) with respect to \( \ln x \), simplifying back to \( \frac{\ln x}{x} \). This confirms our guess. Understanding the chain rule ensures that we are correctly identifying antiderivatives in complex integrals.

Differentiation

Differentiation is the process of finding the derivative, or the rate at which a function is changing at any given point. It's crucial for verifying antiderivatives.
In our step-by-step solution:

• We guessed that \( \frac{(\ln x)^2}{2} \) was the antiderivative for \( \frac{\ln x}{x} \).
• By differentiating \( \frac{(\ln x)^2}{2} \) step by step, we confirmed it returned back to the original integrand.

This step ensures the guess is valid, showing the importance of understanding differentiation techniques not just in solving problems, but also in validating results. Differentiation offers a method to validate the integration steps so that errors can be corrected immediately.

Integrand

The integrand is the function you want to integrate, the main focus of any integral problem. In our exercise, the integrand is \( \frac{\ln x}{x} \). Knowing how to approach an integrand is crucial.

• At first glance, the structure can give hints about its antiderivative. Do you see that both \( \ln x \) and \( \frac{1}{x} \) are related?
• This connection often indicates which function might unroll during differentiation.

By looking at \( \frac{\ln x}{x} \), we compared it with known differentiation rules and structures. Recognizing these relationships is key in guessing potential antiderivatives, and subsequently solving the integral. Understanding the features of the integrand leads the way to accurately integrating complex functions.