Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \int\left(1+e^{x}\right)^{2} e^{x} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{(1 + e^x)^3}{3} + C \).

Step 1: Identify Technique for Integration

Notice that the integral \( \int \left(1+e^{x}\right)^{2} e^{x} \, dx \) looks like it would benefit from a substitution method because \( e^x \) appears in both the exponential factor and the base of the squared expression. This suggests the use of substitution for easier integration.

Step 2: Choose a Suitable Substitution

Let \( u = 1 + e^x \). Then differentiating both sides with respect to \( x \) gives \( du = e^x \, dx \). This allows the substitution of \( e^x \, dx \) with \( du \) in the integral.

Step 3: Substitute and Simplify the Integral

By substituting \( u = 1 + e^x \) and \( du = e^x \, dx \), the integral becomes \( \int u^2 \, du \). This simplification allows us to perform a basic integration on \( u^2 \).

Step 4: Perform the Integration

Integrate \( u^2 \) with respect to \( u \). The integral of \( u^2 \) is \( \frac{u^3}{3} + C \), where \( C \) is the constant of integration.

Step 5: Substitute Back in terms of x

Replace \( u \) with \( 1 + e^x \), returning to the original variable. Thus, the integral becomes \( \frac{(1 + e^x)^3}{3} + C \).

Key Concepts

Substitution Method

The substitution method is a popular technique used to simplify integration problems, especially when direct integration seems complex. This method is akin to reversing the chain rule used in differentiation. By making a smart substitution, you transform the integral into a simpler form that is easier to evaluate. In the given problem, we used the substitution \( u = 1 + e^x \).

To perform substitution, follow these steps:

• Identify the part of the integral: Look for an expression whose derivative also appears in the integral. In this case, \( e^x \) is part of both terms.
• Choose a substitution: Let \( u \) be equal to the identified expression, \( u = 1 + e^x \).
• Find \( du \): Differentiate \( u \) to find \( du \), as \( du = e^x \, dx \). This substitution simplifies the integration variable.
• Rewrite the integral: Substitute \( u \) and \( du \) into the integral to simplify it. The expression becomes \( \int u^2 \, du \).
• Integrate: Solve the simplified integral.

Substitution takes advantage of recognizing complex components that simplify upon substitution, making it a powerful tool in calculus.

Exponential Functions

Exponential functions, denoted as \( f(x) = e^x \), are a fundamental mathematical concept widely used in various fields such as growth models, compound interest calculations, and scientific computations. In our exercise, this function is crucial because it appears both in the base of the squared term and in the multiplication factor of the integral.

Key features of exponential functions include:

• Constant Rate of Growth: Each increase in \( x \) results in multiplication by the base \( e \), meaning the function grows or decays exponentially.
• Derivative and Integral: The unique property of \( e^x \) is that its derivative and integral are both \( e^x \), which simplifies calculations involving differentiation and integration.
• Versatile Applications: From modeling population growth and radioactive decay to financial predictions, exponential functions are integral to many disciplines.

In our specific integral, \( e^x \) facilitates the substitution process, streamlining the evaluation of the integral which otherwise might be convoluted.

Definite and Indefinite Integrals

Integrals can be categorized into two main types: definite and indefinite. Understanding their difference is crucial when conducting integration tasks.

An indefinite integral is a function representing the antiderivative of the original function. It includes a constant of integration, \( C \), reflecting the absence of limits of integration:

• The indefinite integral of \( f(x) \) with respect to \( x \) is written as \( \int f(x) \, dx = F(x) + C \).

In the context of our example, we computed an indefinite integral: \( \int (1 + e^x)^2 e^x \, dx = \frac{(1 + e^x)^3}{3} + C \). This reflects the family of curves obtained by differing \( C \).

A definite integral, on the other hand, has specific limits \( a \) and \( b \), providing the area under the curve \( f(x) \) between these limits:

• The definite integral is represented as \( \int_a^b f(x) \, dx \).
• Evaluating it results in a real number, which is the net area under the curve from \( x = a \) to \( x = b \).

While our problem led to an indefinite integral, recognizing the form one is working with is essential as it alters both the process and the outcome. These concepts lay the foundation for working with integrated functions and their practical applications.