Question

Perform the indicated integrations. $$ \int \frac{e^{x}}{2+e^{x}} d x $$

Short answer

The integral is \( \ln |2 + e^x| + C \).

Step-by-step solution

Identify the Integration Technique

To solve the integral \( \int \frac{e^{x}}{2+e^{x}} \, dx \), we notice that it resembles a form suitable for substitution. Let's use the substitution \( u = 2 + e^x \).

Substitute Variables

Using the substitution \( u = 2 + e^x \), we find the derivative with respect to \( x \): \( du = e^x \, dx \). Thus, \( e^x \, dx = du \). Now, the integral becomes \( \int \frac{1}{u} \, du \).

Integrate the Substituted Function

The integral \( \int \frac{1}{u} \, du \) is a standard integral which equals \( \ln |u| + C \), where \( C \) is the constant of integration.

Substitute Back to Original Variable

Substitute back using \( u = 2 + e^x \) to obtain \( \ln |2 + e^x| + C \).

State the Final Solution

The integral of \( \frac{e^{x}}{2+e^{x}} \) is \( \ln |2 + e^x| + C \), where \( C \) is the constant of integration.

Key concepts

Substitution Method

The substitution method is like changing the perspective to make integration easier. Imagine trying to solve a puzzle: sometimes rearranging the pieces can provide a clearer picture. In integration, substitution works similarly, helping us simplify complex expressions.

Here's how it works generally:

• Select a substitution variable, often called "u". This is chosen based on the inner functions or parts of the integrand that make the problem manageable.
• Find the derivative of the substitution equation to replace the original variable's differential.
• Change the integral to the new variable "u". This often turns a complicated integral into a simpler one.

In our example, we chose to let \( u = 2 + e^x \). This choice transformed the integral into \( \int \frac{1}{u} \, du \), which is much simpler to integrate. After finding the antiderivative, we substitute back the original variables to solve the problem completely.

Exponential Functions

Exponential functions have the form \( e^x \), where \( e \) is a mathematical constant approximately equal to 2.71828. This type of function grows rapidly and appears frequently in calculus problems due to its unique properties.

Some important characteristics of exponential functions include:

• The derivative of \( e^x \) is \( e^x \), which is a key feature making it prevalent in both differential and integral calculus.
• Exponential functions are continuous and smooth, making them relatively straightforward to work with in integration problems.
• They often appear in growth and decay models, such as population growth or radioactive decay, which makes them practical in real-world scenarios.

In the integration problem we worked on, \( e^x \) was part of the function \( 2 + e^x \). By understanding how exponential functions behave, we could manipulate it appropriately to apply the substitution method.

Indefinite Integrals

Indefinite integrals are the opposite of derivatives and are essential in calculus. They represent a family of functions whose derivatives yield the integrand. Think of them as "anti-derivatives."

Here's what you should know about indefinite integrals:

• Unlike definite integrals, indefinite integrals do not have limits, meaning they represent a general form of the area under the curve.
• The result of an indefinite integral includes a "+ C," known as the constant of integration. This constant accounts for all possible versions of a function whose derivative matches the integrand.
• Every differentiation problem has a corresponding indefinite integral problem, serving as a balance between the two operations.

In our exercise, we computed the indefinite integral \( \int \frac{e^{x}}{2+e^{x}} \, dx \) which ultimately became \( \ln |2 + e^x| + C \). This solution gives us all the functions whose derivative is the original integrand.