Problem 39 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(39-56\) $$\int_{-3}^{-2} \frac{d x}{x}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral

The given integral is \(\int_{-3}^{-2} \frac{d x}{x}\). This is a definite integral of the function \(\frac{1}{x}\) from \(-3\) to \(-2\).

Step 2: Recognize the Antiderivative

The antiderivative of \(\frac{1}{x}\) is \(\ln |x|\). This function is known and can be used to evaluate the integral.

Step 3: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, to evaluate the definite integral, we substitute the upper bound and lower bound into the antiderivative: \(\int_{-3}^{-2} \frac{d x}{x} = \ln|-2| - \ln|-3|\).

Step 4: Simplify the Expression

Simplify the expression obtained from substitution: \(\ln 2 - \ln 3\). Use the logarithmic property \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\) to further simplify: \(\ln \left( \frac{2}{3} \right)\).

Step 5: Final Result

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

Key Concepts

Understanding the Antiderivative

When tackling definite integrals, one important step is recognizing the antiderivative of the function you're integrating. An antiderivative, also known as the indefinite integral, is a function whose derivative equals the original function. Consider the function \( \frac{1}{x} \). Its antiderivative is \( \ln |x| \). This means that the derivative of \( \ln |x| \) will give us back \( \frac{1}{x} \).

This familiar concept is crucial because it allows us to relate differentiation and integration. Identifying the correct antiderivative provides the foundation for solving definite integrals. In terms of applications, this process is used in various fields such as physics and economics to solve real-world problems.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links together the processes of differentiation and integration, which are two of the core operations in calculus. This theorem is divided into two parts. The first part tells us how a continuous function has an antiderivative that can be found by integrating the function.

The second part of the theorem is particularly useful for evaluating definite integrals. According to this part, if you have a continuous function \( f \) on the interval \([a, b]\), and \( F \) is an antiderivative of \( f \), then the definite integral of \( f \) from \( a \) to \( b \) is given by \( F(b) - F(a) \).

• This means you simply need to find the antiderivative of the function, evaluate it at the upper and lower bounds, then subtract the two results.
• In our example, for \( \int_{-3}^{-2} \frac{1}{x} \, dx \), the antiderivative was \( \ln |x| \). So, we calculate \( \ln |-2| - \ln |-3| \).