Problem 19 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=e^{-3 x} $$

Step-by-Step Solution

Answer

The general antiderivative of \( f(x) = e^{-3x} \) is \( -\frac{1}{3} e^{-3x} + C \).

Step 1: Identify the Basic Form

The function given is in the form of the exponential function: \( f(x) = e^{kx} \). The formula for the antiderivative in this case is \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \), where \( C \) is the constant of integration.

Step 2: Determine the Constant \( k \)

Here the exponent is \(-3x\), therefore, \( k = -3 \). This means we will use \( k = -3 \) in the antiderivative formula.

Step 3: Apply the Antiderivative Formula

Using the formula \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \), substitute \( k = -3 \) into the formula: \( \int e^{-3x} \, dx = \frac{1}{-3} e^{-3x} + C \).

Step 4: Simplify the Expression

Simplifying \( \frac{1}{-3} e^{-3x} + C \) gives \( -\frac{1}{3} e^{-3x} + C \). This expression represents the general antiderivative of the function.

Key Concepts

Exponential Function

Exponential functions are a fundamental type of mathematical function, with the general form \( f(x) = e^{kx} \). Here, \( e \) represents Euler's number, approximately equal to 2.71828, and \( k \) is a constant that can affect the function's growth or decay rate. These functions are commonly used because of their properties related to growth processes found in nature, finance, and various fields of engineering.

In the given exercise, the function \( f(x) = e^{-3x} \) represents an exponential function undergoing exponential decay, because \( k \) is negative. The negative exponent causes the function to decrease as \( x \) increases. Understanding how exponential functions behave is crucial because they often model real-world phenomena, such as population decay, radioactive decay, and cooling processes.

Constant of Integration

When finding an antiderivative or integral of a function, a constant of integration, denoted usually by \( C \), is always part of the solution. This is because indefinite integrals have an infinite number of solutions that differ by a constant value. Thus, when we say a function has an antiderivative, it means there are many possible antiderivatives, each differing only by this constant.

Including the constant \( C \) helps us accommodate all these potential solutions. For example, the basic antiderivative of the exponential function \( e^{kx} \) is \( \frac{1}{k} e^{kx} + C \). Different values of \( C \) produce different functions, showcasing the infinitely many functions that satisfy the original differential equation. It's important to remember the constant of integration when solving integral problems, as it provides the full set of possible solutions for the indefinite integral.

Integration Formula

The integration formula used to find the antiderivative of an exponential function is crucial for solving many integration problems. The specific formula, \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \), is applied when integrating functions of the form \( e^{kx} \). This process allows us to "undo" the differentiation process, after which we retrieve the function whose derivative was given.

Using this formula involves a few simple steps:

• Identify the constant \( k \) in the exponent.
• Apply the formula by replacing \( k \) with its value in the expression.
• Add the constant of integration \( C \) at the end.

This method simplifies finding antiderivatives for exponential functions, and is fundamental in calculus. Mastering the use of this formula enables solving a variety of problems involving exponential growth and decay functions, making it an essential tool in calculus.