Problem 37 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(37-54\). $$ \int_{-3}^{-2} \frac{d x}{x} $$

Step-by-Step Solution

Answer

The value of the integral is \(\ln \frac{2}{3}\).

Step 1: Identify the Function to Integrate

We are given the integral \( \int_{-3}^{-2} \frac{dx}{x} \). The function to integrate is \( \frac{1}{x} \), which is the standard form for a natural logarithm integral.

Step 2: Apply the Fundamental Theorem of Calculus

The integral \( \int \frac{1}{x} \, dx \) is equal to \( \ln |x| + C \), where \( C \) is the constant of integration. Since we are evaluating a definite integral from \(-3\) to \(-2\), we need to compute \( \ln |-2| - \ln |-3| \).

Step 3: Calculate the Definite Integral

Evaluate the expression \( \ln |-2| - \ln |-3| \). Since negative signs in the absolute value function make both arguments positive, it simplifies to \( \ln 2 - \ln 3 \).

Step 4: Apply the Property of Logarithms

Use the logarithmic property \( \ln a - \ln b = \ln \frac{a}{b} \) to further simplify \( \ln 2 - \ln 3 \) to \( \ln \frac{2}{3} \). This is the value of the definite integral.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a key concept that links differentiation and integration. It shows the relationship between the two main operations in calculus, providing a tool to evaluate definite integrals. It states that if a function is continuous over an interval \([a, b]\), then the integral of a function can be found by using its antiderivative. If \(F(x)\) is an antiderivative of \(f(x)\), the theorem is stated as follows:

• \[ \int_a^b f(x) = F(b) - F(a)\]

In our exercise, the function \(\frac{1}{x}\) is continuous over the given interval \((-3, -2)\) except at the undefined point \(x=0\). The definite integral evaluates the change in the antiderivative \(\ln |x|\) between \(-3\) and \(-2\).
This demonstrates how we apply the theorem to compute the definite integral effectively by considering the properties of the antiderivative function.

Natural Logarithm

The natural logarithm, denoted by \(\ln(x)\), is the logarithm to the base of Euler's number \(e\), and is commonly used in calculus due to its nice properties. In this exercise, the integral \(\int \frac{1}{x} \, dx\) directly relates to \(\ln|x|\). Here’s why the natural logarithm is useful:

• The derivative of the natural logarithm function \(\ln(x)\) is \(\frac{1}{x}\).
• This relationship makes it straightforward to identify that the antiderivative of \(\frac{1}{x}\) is \(\ln|x| + C\).

Since we're dealing with a definite integral, the constant \(C\) cancels out in the expression \(\ln |-2| - \ln |-3|\).
Understanding the properties and role of the natural logarithm helps in grasping the function's transformation during integration.

Properties of Logarithms

Logarithms have unique properties that make them incredibly useful for simplifying expressions, especially in calculus. One such property applied in the solution is the difference property:
\[ \ln a - \ln b = \ln \frac{a}{b} \]This property simplifies the integration result from \(\ln 2 - \ln 3\) to \(\ln \frac{2}{3}\), by taking advantage of the fact that subtraction between logarithms results in a logarithm of a quotient.
The proper use of these properties helps simplify calculations and makes integration solutions more elegant and easy to interpret. Another useful property to remember is:

• \[ \ln(x^n) = n \cdot \ln(x) \]

Using these properties efficiently can significantly ease solving complex problems involving logarithmic integration.