Problem 41 - 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. $$ f(x)=\frac{e^{(a+1) x}}{a} $$

Step-by-Step Solution

Answer

\( \int f(x) \, dx = \frac{1}{a(a+1)} e^{(a+1)x} + C \).

Step 1: Identify the Function and Constant

The given function is \( f(x) = \frac{e^{(a+1) x}}{a} \), where \( a \) is a positive constant. Our task is to find the antiderivative (or indefinite integral) of this function with respect to \( x \).

Step 2: Factor Out the Constant

Since \( \frac{1}{a} \) is a constant with respect to \( x \), we can factor it out of the integral. So we need to integrate \( \frac{1}{a} \int e^{(a+1)x} \, dx \).

Step 3: Integrate the Exponential Function

The antiderivative of \( e^{(a+1)x} \) with respect to \( x \) is \( \frac{1}{a+1}e^{(a+1)x} \), using the formula for integrating exponential functions: \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \), where \( k \) is a constant.

Step 4: Combine the Results and Simplify

Multiply the constant \( \frac{1}{a} \) with the antiderivative from Step 3: \( \frac{1}{a} \times \frac{1}{a+1}e^{(a+1)x} = \frac{1}{a(a+1)}e^{(a+1)x} \).

Step 5: Add the Constant of Integration

Finally, add the constant of integration \( C \) to represent the most general form of the antiderivative: \( \int f(x) \, dx = \frac{1}{a(a+1)} e^{(a+1)x} + C \).

Key Concepts

Indefinite Integrals

When we talk about indefinite integrals, we're essentially looking for a function whose derivative matches the original function.
This means that if we were to differentiate the antiderivative, we'd get our original function back. An indefinite integral does not have upper or lower bounds on the integral sign, and it represents a family of functions.
Every indefinite integral comes with a constant of integration, denoted as 'C', which accounts for the infinite number of potential functions that derivative to the same original function.

• The expression \( \int f(x) \, dx \) represents the antiderivative of the function \( f(x) \).
• The result of an indefinite integral is always in the form of a general expression plus a constant of integration \( C \).

In our exercise, when finding the antiderivative of \( \frac{e^{(a+1)x}}{a} \), we apply the steps of integration to ``undo" the differentiation that created the original function.

Exponential Functions

Exponential functions have a constant base and a variable as an exponent, in the form \( f(x) = a^{x} \), but often in calculus, the base \( a \) is \( e \), the mathematical constant approximately equal to 2.718.
In integration, exponential functions with base \( e \) have a distinct advantage. Their forms are particularly straightforward to integrate and differentiate.
The antiderivative of an exponential function \( e^{kx} \) is \( \frac{1}{k}e^{kx} + C \). This handy formula helps simplify many integration tasks.

• The base \( e \) is special because it creates self-similar functions during differentiation and integration.
• In our problem, \( e^{(a+1)x} \) is the part we need to focus on within the integration process.

We use this property to efficiently find the antiderivative of the exponential expression in the given function, with 'k' being \( a+1 \).

Integration Techniques

Understanding integration techniques is key to successfully finding antiderivatives.One major technique is identifying constants that can be factored out of the integral itself. This streamlines the process and minimizes errors. In problems involving exponential functions, taking advantage of known integration formulas can be a great help.

• First, factor out constants. As in our case, \( \frac{1}{a} \) is pulled out from the integration process, simplifying the function.
• Next, apply integration rules. Using the rule \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \) allows for efficient integration.

Once the antiderivative is calculated, always remember the constant of integration \( C \). This wraps up the integral with a reminder of all possible solutions.In summary, a structured approach helps ensure each step of the process is accurate and complete.