Problem 17 - Applied Calculus

Question

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int e^{3 x} d x $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Integral

We need to evaluate the indefinite integral \( \int e^{3x} \, dx \). This means we are looking for a function whose derivative is \( e^{3x} \).

Step 2: Applying the Substitution Method

We use substitution because the function has a composite form \( e^{u} \). We'll let \( u = 3x \). Then \( du = 3 \, dx \) or \( dx = \frac{du}{3} \).

Step 3: Substituting Variables

Replace \(3x\) with \(u\) in the integral, which gives us \( \int e^u \, \frac{du}{3} \). This simplifies to \( \frac{1}{3} \int e^u \, du \).

Step 4: Integrating the Substituted Function

The integral \( \int e^u \, du \) is simply \( e^u \), as the derivative of \( e^u \) is \( e^u \). Thus, the integral becomes \( \frac{1}{3} e^u \).

Step 5: Back-Substitution

Substitute back \( u = 3x \) into the expression, giving us \( \frac{1}{3} e^{3x} \).

Step 6: Adding the Constant of Integration

Since this is an indefinite integral, we add the constant of integration, \( C \), to our result. Thus, the final answer is \( \frac{1}{3} e^{3x} + C \).

Key Concepts

Substitution Method

The substitution method is a powerful technique for finding indefinite integrals, especially when dealing with composite or more complex functions that are difficult to integrate directly. The basic idea is to transform the original integral into a simpler form, often by changing the variable of integration.When applying the substitution method:

• Identify a part of the integrand (the function inside the integral) that can be substituted with a single variable, often denoted as \( u \).
• Express the differential \( dx \) in terms of \( du \), which typically involves deriving \( u \) with respect to \( x \).
• Substitute the identified variable and its differential into the integral, simplifying the expression.
• After integrating with respect to \( u \), remember to substitute back the original variables to complete the solution.

This method works well with exponential, trigonometric, and polynomial functions, as it simplifies these into more manageable integrals.

Composite Function

A composite function, in the context of integration, involves a function applied within another function. For example, \( e^{3x} \) is a composite function where the exponential function \( e^x \) is applied to the linear function \( 3x \). Composite functions can make integration challenging. Fortunately, the substitution method helps to transform such functions into simpler ones. Essentially, composite functions complicate the derivative of the function you need; hence they often require clever substitutions to unravel. For instance, substituting \( u = 3x \) changes \( e^{3x} \) into \( e^u \), which is easily integrable. As a general rule:

• Look for nested functions where the chain rule is used in differentiation.
• Try substituting the inner function to simplify the integration process.

Using the substitution method can help handle these efficiently, leading to straightforward integral solutions.

Constant of Integration

When working with indefinite integrals, it's crucial to understand the role of the constant of integration, often represented as \( C \). Indefinite integrals represent a family of functions, all differing by a constant. This is because the process of differentiation eliminates any constant values, and thus they cannot be distinctly identified when integrating back.The reason we add a constant of integration:

• To generalize the solution, accounting for any constants lost during differentiation.
• This constant ensures that all possible antiderivatives of the function are considered.
• It is crucial for problems involving initial conditions or specific values, as \( C \) can be determined with extra information.

In sum, remember to always include \( C \) in indefinite integrals to ensure a complete and inclusive solution.