Problem 63 - Applied Calculus

Question

Choose the correct answer. \(\int e^{-1} d x=?\) a. \(\ln |e|+C\) b. \(\frac{1}{0} e^{0}+C\) c. \(e^{-1} x+C\)

Step-by-Step Solution

Answer

c. \( e^{-1} x + C \)

Step 1: Identify the Integral

The integral given is \( \int e^{-1} \, dx \). It involves a constant function because \( e^{-1} \) is a constant. Thus, the integral of a constant \( a \) with respect to \( x \) is \( ax + C \), where \( C \) is the constant of integration.

Step 2: Apply the Formula for Integral of a Constant

Using the integral formula for a constant, we have \( \int e^{-1} \, dx = e^{-1}x + C \). The term \( e^{-1} \) is a constant coefficient, similar to any real number multiplied by \( x \) plus a constant \( C \).

Step 3: Confirm the Correct Option

Compare the result from Step 2, \( e^{-1}x + C \), with the provided options. The correct answer is option c: \( e^{-1}x + C \).

Key Concepts

Constant Function Integration

Integration is remarkably simplified when dealing with constant functions. A constant function remains the same, regardless of the input. In this context, the constant function is represented by \( e^{-1} \). When we integrate a constant, we can apply the straightforward formula:

• Integral of constant \( a \) with respect to \( x \) is \( ax + C \).

Here, \( e^{-1} \) is treated just like any other constant, such as \( 5 \) or \( -3 \). Therefore, the integral \( \int e^{-1} \, dx \) translates perfectly to \( e^{-1}x + C \). It’s an essential tool when solving math problems involving constants because it allows you to bypass complex calculations and directly reach the result by multiplying the constant by the variable of integration.

Definite and Indefinite Integrals

There are two main types of integrals in calculus, known as definite and indefinite integrals. Understanding their differences helps in choosing the correct method for solving integration problems. Here's a quick overview:

• Indefinite Integrals: These are integrals without specific limits of integration. The result of an indefinite integral is a general formula plus a constant \( C \), because integrals represent families of functions. This is the category where our original problem fits in.
• Definite Integrals: In contrast, definite integrals have upper and lower limits, often denoted as \( a \) and \( b \). They provide numerical values rather than function forms and represent the exact area under the curve between those limits.

In solving integration problems, recognizing whether you are dealing with a definite or indefinite integral is crucial, as it influences both the approach and the interpretation of the result.

Basic Integral Formulas

Familiarity with basic integral formulas is a cornerstone of successfully tackling calculus problems. They serve as tools that can simplify complex integrals into manageable operations. Let’s highlight a few key formulas and how they apply:

• Integral of a Constant: \( \int a \, dx = ax + C \): This was used in our example problem to find the integral of the constant \( e^{-1} \).
• Power Rule for Integration: For any rational number \( n \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \).
• Exponential Functions: \( \int e^x \, dx = e^x + C \): Crucial for dealing with expressions involving the base of the natural logarithm.

These basic integral formulas are essential tools. They allow calculus students to efficiently solve a wide range of integration problems by providing a go-to method for different kinds of mathematical expressions encountered in integrals. Mastering these formulas is vital, as it lays the groundwork for more advanced calculus challenges.