Problem 46 - Calculus for Biology and Medicine

Question

In Problems \(41-46\), assume that \(a\) is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{e^{-a x}+e^{a x}}{2 a} $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = \frac{-1}{2a^2} e^{-a x} + \frac{1}{2a^2} e^{a x} + C \).

Step 1: Break Down the Function

The given function is \[ f(x) = \frac{e^{-a x} + e^{a x}}{2 a} \]We need to find the general antiderivative for \( f(x) \). Let's start by breaking down the expression to recognize patterns we can integrate.

Step 2: Split the Function

You can split the terms in the differentiable form: \[ f(x) = \frac{e^{-a x}}{2 a} + \frac{e^{a x}}{2 a} \]This makes it easier to find the antiderivative of each part separately.

Step 3: Antiderivative of the First Term

For the term \( \frac{e^{-a x}}{2 a} \), we find its antiderivative. Note that the antiderivative of \( e^{-a x} \) is \( -\frac{1}{a} e^{-a x} \). Therefore, the antiderivative of \( \frac{e^{-a x}}{2 a} \) is:\[ \frac{-1}{2a^2} e^{-a x} \]

Step 4: Antiderivative of the Second Term

For the term \( \frac{e^{a x}}{2 a} \), recognize that the integrand follows a similar pattern:The antiderivative of \( e^{a x} \) is \( \frac{1}{a} e^{a x} \). Thus, the antiderivative of \( \frac{e^{a x}}{2 a} \) is:\[ \frac{1}{2a^2} e^{a x} \]

Step 5: Combine the Antiderivatives and Add Constant

Now, combine the antiderivatives from Steps 3 and 4:\[ F(x) = \frac{-1}{2a^2} e^{-a x} + \frac{1}{2a^2} e^{a x} + C \]Here, \( C \) is the constant of integration, since antiderivatives are determined up to an arbitrary constant.

Step 6: Verify by Differentiation

To verify, take the derivative of \( F(x) \):Differentiating yields \( \frac{1}{2a} e^{-a x} + \frac{1}{2a} e^{a x} \), which matches the original \( f(x) \), confirming our solution.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that involve exponents with a constant base and a variable exponent. In generic terms, these are usually in the form of \( e^{kx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828, and \( k \) is a constant. These functions have a unique property where their rate of growth or decay at any point is proportional to their value at that point.Understanding Exponential Functions:

• Natural Base: The number \( e \) is widely used as the base for exponential functions in calculus, due to its natural occurrence in growth processes.
• Growth/Decay: When \( k > 0 \), the function grows as \( x \) increases. Conversely, if \( k < 0 \), the function decays.
• Derivatives: A special feature of exponential functions is that their derivatives are proportional to the functions themselves, which is crucial in solving differential equations and integration problems.

Breaking down the given original exercise, recognizing and working with exponential forms is an integral skill. In our specific problem, dealing with both \( e^{-ax} \) and \( e^{ax} \) requires recognizing these forms to integrate them appropriately.

Integration Techniques

Integration is the process of finding the antiderivative of a given function. Various techniques aid in integrating more complex expressions. For exponential functions, some methods include basic antiderivative rules and recognizing patterns.

Basic Integration

When working with exponential functions, it's vital to remember that:

• The antiderivative of \( e^{ax} \) is \( \frac{1}{a}e^{ax} \).
• The antiderivative of \( e^{-ax} \) is \( -\frac{1}{a}e^{-ax} \).

In our exercise, these rules allow us to split the original function into simpler parts and integrate each separately, facilitating the finding of the complete antiderivative.

Working with Coefficients

When dealing with coefficients in exponential integrals, bring them outside of the integral:

• If you have \( \frac{e^{kx}}{c} \), the coefficient \( \frac{1}{c} \) can be factored out.

By splitting and reducing the problem, integration becomes straightforward and intuitive for complex expressions.

Constant of Integration

When finding the antiderivative of a function, a constant usually appears in the final expression. This is known as the constant of integration, denoted by \( C \). The reason behind this is that integration is essentially the reverse of differentiation. Since the derivative of any constant is zero, when moving backwards (i.e., integrating), you introduce a constant to include all potential functions.

Importance in Solutions

• The constant accounts for all vertical shifts of the antiderivative function that maintain the derivative as \( f(x) \).
• Every specific value of \( C \) corresponds to a distinct member of the family of antiderivatives, each differing by a vertical translation.

For the original exercise, we combine the antiderivatives of split terms and include \( C \) because our goal is the general antiderivative, which encompasses all specific solutions for \( F(x) \). Verifying by differentiating the antiderivative can further confirm the accuracy of our solution.