Problem 11 - Calculus Single Variable

Question

Calculate the indefinite integral. $$ \int \exp (e \cdot x) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{e^{e \cdot x}}{e} + C \).

Step 1: Identify the Integral Form

We need to calculate the indefinite integral of the function \( \exp(e \cdot x) \). The integral we are dealing with is of the form \( \int e^{ax} \, dx \), where \( a = e \).

Step 2: Apply the General Indefinite Integral Rule for Exponential Functions

The general formula for integrating an exponential function is \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \( C \) is the constant of integration. In our case, \( a = e \).

Step 3: Substitute and Simplify

By substituting \( a = e \) into the formula, we get: \[ \int e^{e \cdot x} \, dx = \frac{1}{e} e^{e \cdot x} + C \]. This can be simplified to \[ e^{e \cdot x} + C \].

Step 4: Finalize the Result

Therefore, the indefinite integral of \( \exp(e \cdot x) \) is \[ \frac{1}{e} \cdot e^{e \cdot x} + C \], which simplifies to \[ \frac{e^{e \cdot x}}{e} + C \].

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In our integration problem, the base is the mathematical constant \( e \), also known as Euler's number, which is approximately 2.71828. The variable exponent, in this case, is \( e \cdot x \).
Exponential functions have unique properties:

• They grow rapidly. Because the variable is in the exponent, even small increases in the variable can result in large changes in the function's value.
• They have no bounds. As the variable approaches infinity, so does the function.
• They are continuous and differentiable everywhere on the real number line.

Understanding these properties is crucial for integration, as they govern the behavior of the function over the range of integration. In calculus, the notation \( \exp(x) \) is often used to represent \( e^x \), especially when functions are nested or combined, as in our exercise, where \( \exp(e \cdot x) = e^{e \cdot x} \). It's a compact way to display exponential growth.

Integration Techniques

Integration is a fundamental concept in calculus, representing the accumulation of quantities, such as areas under curves. A variety of techniques exists to address different kinds of integrals. In this exercise, the goal was to integrate an exponential function. Here, we employed a specific rule for integrating exponential functions to solve our problem.
For any integral of the form \(\int e^{ax} \, dx\), the following formula is applied: \[\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\]

Applying the Formula

- Identify \( a \) in the expression. For \( e^{e \cdot x} \), \( a \) is indeed \( e \).- Substitute \( a \) into the formula to integrate the exponential function correctly.- Simplify the expression to achieve the result.Indefinite integrals, unlike definite ones, do not have specified limits of integration, meaning that the result includes a constant of integration \( C \). This is necessary because integrating a function results in many related functions differing only by a constant.

Constant of Integration

In any indefinite integral, you'll notice that we add a \( C \) at the end of our result. This is known as the constant of integration. The reason for this step dates back to the Fundamental Theorem of Calculus.
Integration is essentially the inverse operation of differentiation. Whenever we differentiate a function, constants disappear because the derivative of a constant is zero. So, when we integrate, we "retrieve" these constants under the form \( C \), as we cannot know what constant was lost during differentiation.

Importance of the Constant

• It ensures that all possible antiderivatives of a function are accounted for.
• It highlights the "family" of functions that have the same derivative.
• In practical scenarios, the constant can be determined if additional conditions are provided (such as initial conditions in differential equations).

Remember, leaving out the constant \( C \) would mean losing part of the answer's generality. It’s an essential part of correctly solving indefinite integrals, serving as a placeholder for any number that could have been lost during differentiation.