Problem 23 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Which of the following is an antiderivative of \(\ln x ?\) \(\begin{array}{llll}\text { i. } \frac{1}{x} & \text { ii. } x \ln x & \text { iii. } x \ln x-x & \text { iv. } \arctan (\ln x)\end{array}\) (b) Evaluate \(\int_{1}^{6} \ln x d x\).

Step-by-Step Solution

Answer

The antiderivative of \( \ln x \) is \( x \ln x - x \), and the value of \( \int_{1}^{6} \ln x dx \) is \( 6 \ln 6 - 6 + 1 \).

Step 1: Identify the antiderivative of \( \ln x \)

Based on the Fundamental Theorem of Calculus, the derivative of \( x \ln x - x \) is \( \ln x \). Therefore, the antiderivative of \( \ln x \) is \( x \ln x - x \).

Step 2: Check the remaining options

Calculate the derivative of each option using the chain rule for the derivatives. The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \), the derivative of \( x \ln x \) is \( \ln x + 1 \), and the derivative of \( \arctan(\ln x) \) is \( \frac{1}{x(1+(\ln x)^2)} \). These do not match the function \( \ln x \). Therefore, \( x \ln x - x \) is the correct antiderivative.

Step 3: Evaluate the definite integral of \( \ln x \) from 1 to 6

First, identify the antiderivative of \( \ln x \) which is \( x \ln x - x \), then substitute the limits 1 and 6. Therefore, the integral equals: \[ \int_{1}^{6} \ln x dx = [x \ln x - x]_{1}^{6} \] = \( 6 \ln 6 - 6 - (1 \ln 1 - 1) \) = \( 6 \ln 6 - 6 + 1 \).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a bridge between the concepts of differentiation and integration. It is essential to understand both the process of finding derivatives and integrals.

• The first part of the theorem states that if a function is continuous over an interval, then it has an antiderivative on that interval. This means we can 'reverse' differentiation to find integrals.
• The second part helps evaluate definite integrals. It tells us that if we have an antiderivative of a function, we can compute the integral over an interval by evaluating this antiderivative at the endpoints of the interval and subtracting.

In the context of the problem, the antiderivative of \( \ln x \) is needed. By using the theorem, we verify which function differentiates to \( \ln x \). The solution successfully identifies \( x \ln x - x \) as this antiderivative.

Definite Integrals

Definite integrals are a fundamental concept used to compute the area under a curve within given limits. They are used extensively in many scientific fields.

• A definite integral is written as \( \int_{a}^{b} f(x) \, dx \), where \( f(x) \) is the function to integrate, and \( [a, b] \) is the interval of integration.
• To find the value of a definite integral, we use antiderivatives. Specifically, we determine the antiderivative of the function, evaluate it at the upper and lower limits, and subtract the two results: \( F(b) - F(a) \).

In the exercise, finding the definite integral of \( \ln x \) from 1 to 6 involves calculating \( [x \ln x - x]_{1}^{6} \). This shows how definite integrals provide a numerical value representing the accumulated quantity.

Chain Rule for Derivatives

The chain rule is an essential rule in calculus used to differentiate composite functions. Understanding the chain rule is crucial for verifying antiderivatives, as demonstrated in the solution.

• The chain rule formula is \( \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \), meaning we take the derivative of the outer function and multiply it by the derivative of the inner function.
• This rule is particularly useful when a function is composed of another function, like \( \arctan(\ln x) \).

In this exercise, the chain rule helped check derivative calculations for the given options. By applying the rule, it was confirmed that among the alternatives, only the derivative of \( x \ln x - x \) resulted in \( \ln x \), validating it as the correct antiderivative.