Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Use the Exponential Rule to find the indefinite integral. $$ \int 5 e^{2-x} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( 5e^{2-x} \) with respect to \( x \) is \( -5e^{2-x} + C \)

Step 1: Identify the Function

The given function is \( \int 5e^{2-x} dx \). Here, the base of the exponent is Euler's number \( e \), raised to the power of \(2-x\). Notice that the derivative of the exponent \(2-x\) with respect to \( x \) is \( -1 \), which indicates the use of the exponential rule.

Step 2: Apply Exponential Rule

The formula for the integral of \( e^u \) is \( \int e^u du = e^u+C \), where \( C \) is the constant of integration. However, if we have to take the integral of a function in the form \( e^{f(x)} \), where \( f'(x) \) is the derivative of \( f(x) \), we have \( \int e^{f(x)} f'(x) dx = e^{f(x)} + C \). Here, the derivative of \( 2 - x \) is \( -1 \), so we will have to multiply by \( -1 \) to correct this.

Step 3: Perform the Integration

Perform the integration. \( \int 5e^{2-x} dx = -5 \int e^{2-x} d(2-x) = -5e^{2-x} + C \)

Step 4: Simplify the Result

The answer after simplification becomes \( -5e^{2-x} + C \)

Key Concepts

Exponential Rule

The Exponential Rule is a key concept in calculus, especially when dealing with integrals and derivatives involving exponential functions. This rule is essential for solving integrals of the form \( \int e^{f(x)} f'(x) \, dx \). In simple terms:

• If you have an exponential function \( e^{f(x)} \), where \( f(x) \) is any differentiable function, then its derivative \( f'(x) \) is necessary to directly integrate this function using the exponential rule.
• The general formula is: \( \int e^{f(x)} f'(x) \, dx = e^{f(x)} + C \), where \( C \) is the constant of integration.

When applying the exponential rule, it is crucial to check if the differential \( f'(x) \, dx \) is present. If it's not, adjustments can be made, such as multiplying and dividing to fit the rule's requirements.

Constant of Integration

In calculus, the constant of integration \( C \) is an arbitrary constant added to the function's antiderivative. It arises because the process of differentiation removes any constant we have added.

• When integrating, the constant of integration is critical because an indefinite integral represents a family of functions.
• Consider the integral \( \int f(x) \, dx = F(x) + C \). Here, \( F(x) \) is the antiderivative, and \( C \) represents any constant value.

Remember, the presence of \( C \) ensures that all possible primitive functions are captured, accounting for solutions differing by a constant.

Derivative

The derivative is a fundamental concept in calculus, representing the rate at which a function is changing at any point. It is defined as the limit:\[\frac{d}{dx} f(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}\]Derivatives are crucial in understanding the behavior of functions:

• They help us find rates of change, such as speed or growth rate.
• They are used to determine the function's slope at any given point.

Calculating the derivative is the reverse process of finding an integral. When integrating a function, you're essentially finding the original function from its rate of change.

Euler's Number

Euler's number, denoted as \( e \), is a mathematical constant approximately equal to 2.71828, and it is the base of the natural logarithm. Euler's number has unique properties:

• It is an irrational number, meaning it cannot be expressed as a simple fraction.
• The function \( e^x \) is its own derivative and integral, demonstrating its unique nature in calculus.

Euler's number is named after the Swiss mathematician Leonhard Euler. Its natural occurrence in various mathematical contexts, such as compound interest, wave functions, and probability theory, highlights its importance across scientific disciplines.