Problem 62 - Applied Calculus

Question

Choose the correct answer. \(\int e^{3} d x=?\) a. \(\frac{1}{4} e^{4}+C\) b. \(e^{3}+C\) c. \(e^{3} x+C\)

Step-by-Step Solution

Answer

(c) \( e^3 x + C \) is the correct answer.

Step 1: Understand the Problem

The problem is to integrate the function \( e^3 \) with respect to \( x \), which involves finding the antiderivative.

Step 2: Identify Constant Multiplication in Integration

Since the function \( e^3 \) is a constant (because \( e^3 \) is a constant number and not a function of \( x \)), its integration involves multiplying by the variable of integration, \( x \).

Step 3: Apply Integration Rule for Constants

For constants, the integration rule is \( \int a \cdot dx = a \cdot x + C \), where \( a \) is a constant. Here, \( a = e^3 \).

Step 4: Perform the Integration

Applying the integration rule to \( e^3 \), we get: \( \int e^3 \cdot dx = e^3 x + C \).

Step 5: Select the Correct Answer

From the options given, choice (c), \( e^3 x + C \), matches our solution.

Key Concepts

Antiderivative

The term "antiderivative" refers to one of the fundamental concepts in calculus, which involves finding a function whose derivative matches a given function. Consider the example of determining the antiderivative of a constant such as \(e^3\). Instead of looking for a rate of change, we seek the original function before differentiation.
When finding an antiderivative, the goal is to reverse the differentiation process. In cases involving constants, like \(e^3\), the value itself does not change with respect to \(x\), distinguishing constants from variable-dependent functions.
It is worth noting that finding an antiderivative is part of the broader process known as integration. Integration collects or aggregates values over an interval, effectively 'adding up' instances to find cumulative results. This makes it essential in applications such as calculating areas under curves or accumulated quantities in physics and engineering.

Constant Integration

A fundamental rule in integration is the constant rule. This rule states that the integral of a constant with respect to a variable results in the constant multiplied by the variable. This is typically expressed as \(\int a \, dx = a \cdot x + C\), where \(a\) is a constant.
In the given problem involving \(e^3\), because \(e^3\) is a constant, integrating it with respect to \(x\) results in \(e^3 x + C\).

• This formula applies universally to any real constant \(a\).
• \(C\) represents the "constant of integration," accounting for any constant that might have existed before differentiation.

When integrating constants, always include \(C\) to fully capture the family of antiderivatives that result from indefinite integration.

Integration Rules

Integration rules provide a systematic way to find antiderivatives of functions. Commonly used rules include those for polynomials, exponential functions, and constants.
In the context of this problem, the relevant rule is for integrating constants, which follows the pattern \(\int a \, dx = a \cdot x + C\). This rule simply scales the constant by the variable of integration.
Several integration techniques and rules exist:

• Power Rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), for \(n eq -1\).
• Exponential Rule: \(\int e^x \, dx = e^x + C\).
• Trigonometric Integrals: Utilize identities to integrate functions involving sin, cos, etc.

Understanding these rules is vital for successfully tackling a variety of calculus problems and ensuring the correct application to each unique function.