Problem 57 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int e^{2 x} d x $$

Step-by-Step Solution

Answer

\( \int e^{2x} \, dx = \frac{1}{2} e^{2x} + C \)

Step 1: Recognize the Integral Form

The integral given is \( \int e^{2x} \, dx \). This is an exponential function, and we need to use the standard integration rule for exponential functions. The general form for integration of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \), where \( a \) is a constant and \( C \) is the constant of integration.

Step 2: Identify Constants and Apply the Formula

In our integral \( \int e^{2x} \, dx \), we identify \( a = 2 \). Applying the formula, the integral becomes \( \frac{1}{2} e^{2x} + C \). Here, \( C \) is the constant of integration which you add since this is an indefinite integral. This ensures all possible antiderivatives are accounted for.

Key Concepts

Understanding the Exponential Function

An exponential function is a mathematical expression in which a variable appears in the exponent. The most common exponential function is the natural exponential function, which is written as \( e^x \). Here, \( e \) is a mathematical constant approximately equal to 2.71828. It's the base of the natural logarithm and is often encountered in growth and decay problems. Exponential functions, like \( e^{2x} \), play a crucial role in various branches of mathematics, including calculus and differential equations. They model many phenomenons including population growth, radioactive decay, and interest calculations, making them essential for both theoretical and real-world applications. When working with exponential functions in calculus, understanding their growth behavior and how they can be manipulated algebraically is crucial. For instance, the exponential function \( e^{2x} \) grows more rapidly than \( e^x \), as each increase in \( x \) is magnified by the factor 2. This rapid growth rate is a characteristic property of exponential functions.

Applying the Integration Rule

Integration is the reverse process of differentiation. For exponential functions specifically, the integration rule can be directly applied given their straightforward derivative properties. The standard rule for integrating \( e^{ax} \) is \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \).

• The constant \( a \) in the exponent helps to determine the factor \( \frac{1}{a} \) that's needed outside the exponential expression. This accounts for the effect that scaling the variable \( x \) by \( a \) has on the rate of change.
• Using this rule streamlines the integration process, turning the task into a simple application once \( a \) is identified from the function.

In the original problem \( \int e^{2x} \, dx \), we identify \( a = 2 \). According to the rule, the anti-derivative is \( \frac{1}{2} e^{2x} + C \). This formula not only provides the solution but also suggests a methodical and time-saving approach for similar problems involving exponential functions.

The Role of the Constant of Integration

When you're working on an indefinite integral, you come across something called the constant of integration, represented by \( C \). This constant is essential because when you integrate a function, you find a family of functions that differ by a constant.

• The constant of integration accounts for all possible vertical shifts of the function \( e^{2x} \). Each possible value of \( C \) represents a different antiderivative.
• No initial condition is specified, meaning without \( C \), we can't uniquely identify the original function from its derivative.
• Including \( C \) ensures that the solution to the indefinite integral reflects every possible function that could have yielded the integrand through differentiation.

This aspect of integration highlights the difference between definite and indefinite integrals. Definite integrals calculate the exact area under a curve over a specific interval, whereas indefinite integrals sketch out a broader landscape of potential functions.