Problem 30 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int_{1}^{2}\left(\frac{1}{x}-e^{-x}\right) d x$$

Step-by-Step Solution

Answer

The integral evaluates to \( \ln 2 - e^{-1} + e^{-2} \).

Step 1: Identify the Integral Components

The given integral is \( \int_{1}^{2} \left( \frac{1}{x} - e^{-x} \right) dx \). This can be separated into two integrals: \( \int_{1}^{2} \frac{1}{x} \, dx \) and \( \int_{1}^{2} e^{-x} \, dx \).

Step 2: Evaluate the First Integral

The first integral is \( \int_{1}^{2} \frac{1}{x} \, dx \). The antiderivative of \( \frac{1}{x} \) is \( \ln|x| \). Thus, \( \int_{1}^{2} \frac{1}{x} \, dx = \ln|x| \bigg|_1^2 = \ln 2 - \ln 1 = \ln 2 \).

Step 3: Evaluate the Second Integral

The second integral is \( \int_{1}^{2} e^{-x} \, dx \). The antiderivative of \( e^{-x} \) is \( -e^{-x} \). Thus, \( \int_{1}^{2} e^{-x} \, dx = -e^{-x} \bigg|_1^2 = -e^{-2} + e^{-1} = e^{-1} - e^{-2} \).

Step 4: Combine the Results

Substitute the results from Steps 2 and 3 into the original integral: \( \int_{1}^{2} \left( \frac{1}{x} - e^{-x} \right) dx = \ln 2 - (e^{-1} - e^{-2}) \). Simplifying gives \( \ln 2 - e^{-1} + e^{-2} \).

Key Concepts

Definite Integrals

In calculus, definite integrals are a core concept that help determine the total accumulation of a function over a specified interval. A definite integral is expressed as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the boundaries of the interval. The definite integral calculates the area under the curve of the function \( f(x) \) from \( x = a \) to \( x = b \). This makes them very useful for finding total quantities related to rate changes, such as displacement from velocity.

• The integral symbol \( \int \) denotes the integration process.
• \( f(x) \) is the function being integrated.
• The \( dx \) term signifies the variable of integration.

This process involves finding the antiderivative of the function and then evaluating it at the upper and lower bounds of the interval. Finally, subtract the value at \( a \) from the value at \( b \) to determine the area or accumulated quantity.

Antiderivatives

Antiderivatives, also known as indefinite integrals, represent the process of reversing differentiation. When you find an antiderivative, you're essentially identifying a function whose derivative is the given function. This is crucial for understanding definite integrals, as they rely on evaluating antiderivatives at specific bounds. For example, the antiderivative of \( \frac{1}{x} \) is \( \ln|x| \), a fundamental integral result.
For definite integrals like \( \int_{1}^{2} \frac{1}{x} \, dx \), the antiderivative \( \ln|x| \) is used. Evaluating this antiderivative from 1 to 2 results in \( \ln 2 - \ln 1 = \ln 2 \).

• Finding an antiderivative is the reverse of differentiation.
• Each basic function has standard antiderivatives.
• Constant terms may appear due to indefinite boundaries.

Antiderivatives make it possible to compute the exact value of the definite integral by substituting the interval's bounds into the found antiderivative.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent, such as \( e^x \). They frequently appear in integrals due to their unique properties, like the constant rate of growth. When dealing with exponential functions in integrals, the antiderivatives often involve exponential terms as well.

• The function \( e^x \) differentiates to itself, thus \( \int e^x \, dx = e^x + C \).
• For functions like \( e^{-x} \), the antiderivative involves a negative exponent: \( \int e^{-x} \, dx = -e^{-x} + C \).

In our exercise, finding the antiderivative of \( e^{-x} \) was key to solving the integral. After computing the antiderivative, it was evaluated over the specified interval \( 1 \) to \( 2 \). This resulted in \( -e^{-2} + e^{-1} \), which when combined with the other component of the integral, provided the solution. Understanding the behavior and integration of exponential functions is vital in calculus as they describe many natural phenomena and processes.