Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int e^{-0.02 x} d x$$

Step-by-Step Solution

Answer

The indefinite integral of the given function is \(-50e^{-0.02x} + C\).

Step 1: Identify the function's form

In this exercise, we are given the function \(e^{-0.02x}\) to integrate. Notice that the function is of the form \(e^{ax}\), where \(a=-0.02\).

Step 2: Apply the exponential integration formula

To find the indefinite integral for a function of the form \(\int e^{ax} dx\), we use the following formula: \[\int e^{ax} dx = \frac{e^{ax}}{a} + C\] where 'a' is a constant, and 'C' represents the constant of integration.

Step 3: Use the formula to find the indefinite integral

Plugging in the values for our given function, we have: \(\int e^{-0.02x} dx = \frac{e^{-0.02x}}{-0.02} + C\)

Step 4: Simplify the resulting expression

To simplify the expression, we can write the final answer as: \(-50e^{-0.02x} + C\) So, the indefinite integral of the given function is \(-50e^{-0.02x} + C\).

Key Concepts

Exponential Functions

Exponential functions are a key concept in mathematics, especially in calculus. These functions are written in the form \(e^{ax}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.718, and \(a\) is a constant coefficient.

Exponential functions are unique because they have a constant rate of growth or decay, which is why they appear often in both mathematics and real-world applications, such as population growth, radioactive decay, and interest calculations. The derivative and integral of exponential functions remain exponential, making them both simple and powerful for mathematical operations.

• The function \(e^{0}\) is equal to 1, which is an important basic property.
• For any real number \(x\), the function \(e^x\) is always positive.
• Exponential functions grow faster than polynomial functions as \(x\) increases.

Understanding exponential functions is crucial for mastering integration techniques and other calculus concepts.

Integration Techniques

Integration techniques are foundational skills in calculus that allow us to find antiderivatives, or indefinite integrals, of functions. The goal of integration is to reverse the process of differentiation. To integrate exponential functions, we often use specific formulas or rules that simplify the process.

For exponential functions of the form \(\int e^{ax} dx\), the integration technique involves using the formula:\[\int e^{ax} dx = \frac{e^{ax}}{a} + C\]where \(a\) is a constant and \(C\) is the constant of integration. This formula helps us find the indefinite integral easily.

• When integrating, always check whether the function can be simplified or rewritten to fit the known formulas.
• The constant \(\frac{1}{a}\) appears in the solution because of the chain rule used during differentiation. It's crucial to multiply by this constant to properly adjust the formula during integration.
• The constant of integration, \(C\), represents an infinite number of possible constants related to all the antiderivatives of the function.

Mastery of these techniques is important for solving more complex calculus problems.

Calculus Concepts

Calculus is a branch of mathematics focusing on change and motion, utilizing derivatives and integrals as its main tools. Understanding these concepts is essential for tackling various problems involving continuous change.

The indefinite integral is a fundamental aspect of calculus, representing a family of functions, or antiderivatives. When we integrate a function, we're looking for a function whose derivative matches the integrand.

• The process of finding the indefinite integral, or antiderivative, does not specify initial conditions, hence the addition of the constant \(C\) at the end of the integral expression.
• In contrast to definite integrals, indefinite integrals do not calculate a numerical value but rather provide a function that describes a particular situation.
• Understanding the relationship between differentiation and integration—often called the Fundamental Theorem of Calculus—helps clarify how they are inverse processes.

Grasping these core calculus concepts allows students to see the bigger picture in problem-solving and better understand the applications of these techniques in real-world scenarios.