Problem 59 - Applied Calculus

Question

\(59-63 .\) Choose the correct answer. \(\int e^{x} d x=?\) a. \(\frac{1}{x+1} e^{x+1}+C\) b. \(e^{x}+C\) c. \(e^{x} x+C\)

Step-by-Step Solution

Answer

The correct answer is b: \( e^{x} + C \).

Step 1: Understand the Integral Expression

We are given the integral \( \int e^{x} \, dx \). We need to find the antiderivative of the exponential function \( e^x \).

Step 2: Recall Derivative Rule

The integral of an exponential function \( e^x \) is one of the basic integration rules. Recall that the derivative of \( e^x \) is \( e^x \), which implies that the integral will reverse this process.

Step 3: Find the Antiderivative

Since the derivative of \( e^x \) is \( e^x \), the integral of \( e^x \) must be \( e^x \). Therefore, the antiderivative of \( e^x \) is \( e^x + C \), where \( C \) is the constant of integration.

Step 4: Choose the Correct Answer

Comparing our result with the options given: - Option a: \( \frac{1}{x+1} e^{x+1}+C \) is incorrect, it relates to another integration.- Option b: \( e^{x}+C \) matches our result.- Option c: \( e^{x} x+C \) involves an additional variable term, making it incorrect.Thus, the correct answer is b.

Key Concepts

Integral of Exponential Function

An integral represents the area under a curve or function. When we talk about the integral of an exponential function like \( e^x \), we are looking for an antiderivative, or a function whose derivative returns us to the original function. This specific task is much simpler because the exponential function \( e^x \) is unique.
The beauty of integrating \( e^x \) lies in the fact that it's one of the rare functions where the derivative and integral are identical. For the function \( e^x \):

• The derivative of \( e^x \) is \( e^x \).
• This means when you integrate \( e^x \), you also get \( e^x \) as the result.

So, the integral of \( e^x \) is simply \( e^x + C \). This comes in handy for solving calculus problems quickly, as the exponential function appears frequently across various mathematical disciplines.

Antiderivative

The concept of an antiderivative is crucial in calculus. An antiderivative is essentially the reverse process of taking a derivative. It is a function that, when differentiated, results in the original function that you started with.
Let's consider the function \( e^x \). We know that:

• The derivative of \( e^x \) is \( e^x \).
• This implies that for finding an antiderivative of \( e^x \), we look for a function whose derivative is \( e^x \).

Therefore, the antiderivative of \( e^x \) is itself, \( e^x \). It's essential to recognize this property to solve integration problems efficiently or when working through differential equations.

Constant of Integration

Every time you integrate a function, you must consider the constant of integration, denoted by \( + C \). But why is this constant so important? Let's break it down.When finding the antiderivative, there isn't a single function that can be the solution, but rather a family of functions. Consider:

• The derivative operation loses any constant value in differentiation.
• As a result, when reversing the operation through integration, any constant that was present originally is lost. In simple terms, integration is blind to constants.

This is where the constant \( + C \) comes in. It represents all possible constant values that the original function might have had. Therefore, the integral of \( e^x \) is written as \( e^x + C \), accommodating any constant that could have been differentiated into nothing, ensuring all possible original functions are considered.