Problem 59 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int 3 e^{-x} d x $$

Step-by-Step Solution

Answer

The integral is \(-3e^{-x} + C\).

Step 1: Identify the Integral Type

The integral \( \int 3 e^{-x} \, dx \) is an indefinite integral involving an exponential function. The integral of \( e^{ax} \) is \( \frac{1}{a} e^{ax} \) plus a constant of integration, where \( a \) is a constant.

Step 2: Recognize the Constant Multiplier

The given integral has a constant multiplier of 3. When integrating, we can factor out this constant, so our integral becomes \( 3 \int e^{-x} \, dx \).

Step 3: Integrate the Exponential Function

We need to integrate \( e^{-x} \). Using the formula for the integral of an exponential function \( e^{ax} \), we have \( \int e^{-x} \, dx = -e^{-x} + C \), where \( C \) is the constant of integration.

Step 4: Multiply the Result by the Constant

Now, multiply the result from Step 3 by the previously factored-out constant \( 3 \). Thus, the integral becomes \( 3 \times (-e^{-x}) + C \), simplifying to \( -3e^{-x} + C \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base, often denoted as \( e \) (approximately 2.718), is raised to a variable exponent. These functions appear frequently in calculus due to their unique properties, especially their growth or decay behavior. The base \( e \) is preferred because of its natural characteristics in mathematical calculations, often simplifying differentiation and integration processes.

When working with indefinite integrals involving exponential functions, it is essential to understand how these functions behave. For instance:

• The derivative of \( e^x \) is \( e^x \), which means that exponential functions are the only functions that remain unchanged when differentiated.
• For exponential decay, characterized by a negative exponent, such as \( e^{-x} \), the function decreases as \( x \) increases.

These properties make exponential functions a core part of solving problems in calculus, physics, and various other fields.

Integration Techniques

Understanding integration techniques is crucial for solving indefinite integrals. The process involves finding the antiderivative of a function, which is a function whose derivative is the original function. Here's a step-by-step to tackle exponential integrals:

• **Recognize the function form**: Identifying that the function is exponential, such as \( e^{-x} \), helps to apply the correct technique.
• **Use constant factors**: When a constant is present, such as 3 in this example, it can be factored out of the integral, allowing you to focus on integrating the exponential function itself.
• **Apply integration rules**: Use the rule that the integral of \( e^{ax} \) is \( \frac{1}{a} e^{ax} \). For \( e^{-x} \), this becomes \( -e^{-x} \) since \( a = -1 \).

These steps guide the integration of various functions, making complex problems more manageable and serve as a framework for tackling different types of functions.

Constant of Integration

The constant of integration, often denoted as \( C \), is a vital aspect of indefinite integrals. When we integrate a function, we find all possible antiderivatives, since integration is essentially the inverse of differentiation.Because the derivative of a constant is zero, the process of integration involves adding this arbitrary constant \( C \) to the integral result. This expresses the family of all possible functions that could have the initial function as a derivative.

Consider when integrating \( e^{-x} \), we find:

• The general integral is \( -e^{-x} + C \). This represents all functions that differentiate to \( -e^{-x} \).
• The constant \( C \) provides flexibility to match specific initial conditions in real-world problems.

Thus, the constant of integration is not just a mathematical formality; it's crucial for finding specific solutions in applied scenarios.