Problem 37 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int_{-3}^{-2} \frac{d x}{x}$$

Step-by-Step Solution

Answer

The integral evaluates to \( \ln{\left(\frac{2}{3}\right)} \).

Step 1: Understanding the Integral

The given integral is \( \int_{-3}^{-2} \frac{dx}{x} \). This is a definite integral with limits from \(-3\) to \(-2\). The function inside the integral is \( \frac{1}{x} \), which is the derivative of \( \ln{|x|} \). Hence, the integral is set up to find the natural logarithm's change between these two limits.

Step 2: Find the Antiderivative

The antiderivative of \( \frac{1}{x} \) is \( \ln{|x|} + C \), where \( C \) is the constant of integration. Therefore, \( \int \frac{dx}{x} = \ln{|x|} + C \). For a definite integral, the constant \( C \) isn't necessary as it cancels out.

Step 3: Evaluate the Definite Integral

To evaluate \( \int_{-3}^{-2} \frac{dx}{x} \), substitute the upper and lower limits into the antiderivative. So, the evaluation becomes \( \ln{|x|} \bigg|_{-3}^{-2} = \ln{|-2|} - \ln{|-3|} \).

Step 4: Simplify the Expression

Now plug in the values: \( \ln{|-2|} = \ln{2} \) and \( \ln{|-3|} = \ln{3} \). So the expression simplifies to \( \ln{2} - \ln{3} \).

Step 5: Final Calculation

The expression can be combined using logarithm properties: \( \ln{2} - \ln{3} = \ln{\left(\frac{2}{3}\right)} \). This is the final value of the definite integral.

Key Concepts

Antiderivative

When dealing with integrals, one of the first steps often involves finding the antiderivative, especially for indefinite integrals. An antiderivative is a function whose derivative is the original function given. This concept is crucial in calculus because it allows us to "reverse" differentiation to find an expression from its rate of change.

In the exercise, the function inside the integral is \( \frac{1}{x} \). The antiderivative of this function is \( \ln{|x|} + C \), where \( C \) represents the constant of integration. However, when evaluating definite integrals, we don't need to worry about this constant, since it will cancel out when applying the limits of integration.

• Antiderivative: A function whose derivative gives the original function.
• Neglect the constant \( C \) in definite integrals, as it cancels out.

Natural Logarithm

The natural logarithm is an important function in mathematics, often represented as \( \ln{x} \). It has a base of \( e \), a mathematical constant approximately equal to 2.71828. The natural logarithm function helps in solving complex problems involving exponential growth and decay due to its unique properties.

In the context of this exercise, the integral \( \int \frac{dx}{x} \) results in the natural logarithm \( \ln{|x|} \). This shows that the function \( \frac{1}{x} \) is the derivative of \( \ln{|x|} \), connecting the integral and natural logarithm in a meaningful way.

• Natural Logarithm: Denoted as \( \ln{x} \), with base \( e \).
• Important for solving exponential problems.

Properties of Logarithms

Logarithms have several useful properties that simplify calculations, particularly when dealing with the final results of integrals. In this exercise, the final step involves simplifying the expression \( \ln{2} - \ln{3} \) using logarithmic properties.

One critical property of logarithms is: \( \ln{a} - \ln{b} = \ln{\left(\frac{a}{b}\right)} \). This property allows us to combine or simplify expressions involving subtraction of logarithms easily. Applying this to the exercise, \( \ln{2} - \ln{3} \) becomes \( \ln{\left(\frac{2}{3}\right)} \).

• Subtraction: \( \ln{a} - \ln{b} = \ln{\left(\frac{a}{b}\right)} \).
• Properties simplify the process of dealing with complex expressions.