Problem 9 - Applied Calculus

Question

Find each indefinite integral. \(\int e^{-0.5 x} d x\)

Step-by-Step Solution

Answer

The indefinite integral is \(-2e^{-0.5x} + C\).

Step 1: Recognize the Integral Form

The integral presented is \ \ \( \int e^{-0.5x} \, dx \). This is a basic exponential function, and the form closely resembles the integral of \( e^{ax} \), which is \( \frac{1}{a} e^{ax} \), plus a constant of integration.

Step 2: Identify the Coefficient 'a'

In the function \( e^{-0.5x} \), the coefficient \( a \) of \( x \) is -0.5. Recognizing this coefficient is crucial to applying the integral rule for exponential functions.

Step 3: Apply the Integral Rule for Exponentials

Using the rule \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \( a = -0.5 \), substitute \( a \) into the formula: \ \ \( \int e^{-0.5x} \, dx = \frac{1}{-0.5} e^{-0.5x} + C \).

Step 4: Simplify the Expression

The expression \( \frac{1}{-0.5} \) simplifies to -2. Substitute this into the equation: \ \ \( -2 e^{-0.5x} + C \). Thus, the indefinite integral is \( -2 e^{-0.5x} + C \).

Key Concepts

Integration Techniques

Understanding integration techniques is like unfolding a toolkit packed with methods for solving various types of integrals. When approaching an indefinite integral, the objective is to determine a function whose derivative results in the original integrand. Indefinite integrals are generally represented with the symbol \( \int \) followed by the function to be integrated, ending with \( dx \) indicating the variable of integration.

Here are some common techniques useful to remember while tackling integrals:

• Basic Antiderivatives: Recognize basic forms like \( \int x^n \, dx \), which results in \( \frac{x^{n+1}}{n+1} + C \).
• Substitution: This method is valuable when the integral includes a composite function. It involves changing variables to simplify the integration process.
• Integration by Parts: Applicable for products of functions, based on the rule \( \int u \, dv = uv - \int v \, du \).
• Recognizing Patterns: Being able to spot common patterns such as exponentials and trigonometric functions can simplify the work significantly.

Every student should aim to practice these techniques, as they are foundational in solving calculus problems efficiently.

Exponential Functions

Exponential functions are a cornerstone in calculus due to their unique properties and prevalence in both real-world applications and mathematical modeling. An exponential function has the form \( e^{ax} \), where \( e \) is the base of natural logarithms approximately equal to 2.718, and \( a \) is a constant.

Key characteristics of exponential functions include:

• Rapid growth or decay depending on whether \( a \) is positive or negative.
• Always positive, as the output of any exponential function is greater than zero.
• The derivative of \( e^{ax} \) remains an exponential function: \( \frac{d}{dx} e^{ax} = a e^{ax} \). This feature makes integration straightforward.

When integrating an exponential function like \( e^{-0.5x} \), one can apply the rule \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \). The key step is identifying the constant \( a \), as seen in the original problem where \( a = -0.5 \). Mastery of these rules can significantly expedite solving exponential integrals.

Calculus Problem Solving

Tackling calculus problems may seem daunting, but it becomes manageable with a systematic approach. The essence of problem-solving lies in:

• Understanding the Problem: Analyze the integral to determine its type. Is it an exponential, polynomial, or something else?
• Choosing the Right Technique: Apply the most fitting integration technique. In our example, the exponential integral rule was the most appropriate choice.
• Computing the Solution: Execute the integration by accurately applying mathematical rules and simplifying where possible.
• Checking Work: Always differentiate the antiderivative obtained to verify it corresponds to the original integrand.

It helps to write everything down clearly. Often, the visual representation of variables and constants contributes to uncovering hidden patterns or errors. Patience and practice play a key role in refining calculus problem-solving skills, leading to more intuitive understanding and faster resolution of complex problems.