Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \ln 3 x d x $$

Step-by-Step Solution

Answer

The integral of \( \ln 3x \) is \( x (\ln 3x - 1) \).

Step 1: Identify integral type

We are asked to evaluate the integral of \( \ln 3x \), which is a natural logarithm. Here we use a simple substitution method without integration by parts.

Step 2: Apply substitution

Let \( u = 3x \). Therefore, \( du = 3dx \). We then rewrite the integral in terms of \( u \).

Step 3: Solve the integral

Once the integral has been rewritten, it can now be solved using the rules for integrating the natural logarithm. \( \int \ln u (du/3) = (1/3) \int \ln u du \). The integral of \( \ln u \) is \( u(\ln u - 1) \).

Step 4: Substitution back into original variable

The final step is to substitute back for \( x \), the original variable of integration, giving the final answer: \( (1/3) \times 3x (\ln 3x - 1) \).

Key Concepts

Natural Logarithm Integration

Integrating natural logarithms can seem tricky at first, especially because they mix algebraic and transcendental functions. The natural logarithm, denoted as \( \ln(x) \), is a fundamental mathematical function. When we talk about the integration of \( \ln(x) \), we are essentially looking for a function whose derivative is the natural logarithm. This is more complex than standard polynomials but can be managed by some specific techniques.

The integral for a natural logarithm \( \ln(u) \) follows a particular rule:

• The integral of \( \ln(u) \) with respect to \( u \) is \( u(\ln(u) - 1) + C \).

This rule induces a simple, step-by-step process for integrating functions that involve \( \ln(x) \). It's crucial to remember these integration rules to handle more complex integrals in calculus effectively.

Substitution Method

The substitution method is a handy technique for solving integrals that might look complex at first glance. It's like changing the "variables" in the equation to make it easier to solve. In the example of finding the indefinite integral of \( \ln(3x) \), substitution greatly simplifies the process.

Here's how the substitution method works:

• Choose a substitution that simplifies the integral. For \( \ln(3x) \), let \( u = 3x \).
• Find the derivative of \( u \) with respect to \( x \), hence \( du = 3dx \).
• Rewrite the integral using \( u \) in place of \( 3x \) and adjust accordingly. This might look like \( \int \ln(u) (du/3) \).
• Once you have the integral rewritten, solve it using known techniques, and then "substitute back" to the original variable at the end.

The key idea is that substitution transforms a tough integral into one that's easier to handle. This approach is often the first step in tackling many integration problems.

Integration Techniques

When dealing with integrals, especially indefinite integrals, a variety of techniques can come into play. The choice of technique, whether substitution, parts, or straightforward rules, depends largely on the form of the integral you're facing.

Here are several common techniques:

• Substitution Rule: Often used for integrals with composite functions, making them easier to integrate by simplifying the variable.
• Integration by Parts: Not applicable in our specific exercise, is useful when the product of two functions is involved.
• Standard Formulas: Many integrals can be resolved using fundamental calculus formulas, such as the integral of \( \ln(u) \).

Each technique offers a pathway to simplify and solve integrals effectively. Understanding when and how to apply these techniques can turn calculus from a daunting subject into a manageable and even enjoyable task. In our exercise, the use of substitution simplified the integral of \( \ln(3x) \) without requiring more complex methods like integration by parts.