Problem 31 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int \frac{\ln x^{2}}{x} d x \text { (Hint: Use the properties of logarithms.) } $$

Step-by-Step Solution

Answer

\((\ln x)^2 + C\) where \( C \) is the constant of integration.

Step 1: Simplify the Integrand

To simplify the integrand \( \frac{\ln x^2}{x} \), use the logarithm property: \( \ln x^2 = 2 \ln x \). Thus, the integrand becomes: \( \frac{2 \ln x}{x} \).

Step 2: Set Up the Integral

Rewrite the integral using the simplified form from Step 1:\[ \int \frac{2 \ln x}{x} \, dx = 2 \int \frac{\ln x}{x} \, dx \]

Step 3: Use Integration by Parts

Apply integration by parts to \( 2 \int \frac{\ln x}{x} \, dx \). Choose \( u = \ln x \) (hence, \( du = \frac{1}{x} \, dx \)) and \( dv = \frac{1}{x} \, dx \) (hence, \( v = \ln x \)).Integration by parts formula: \( \int u \, dv = uv - \int v \, du \).Substitute into the formula:\[ uv = (\ln x)^2, \quad \int v \, du = \int \ln x \, \frac{1}{x} \, dx \]

Step 4: Solve the Integral

Substituting back, we get:\[ 2 \left[ \frac{(\ln x)^2}{2} - \int \ln x \, \frac{1}{x} \, dx \right] = (\ln x)^2 - \int \ln x \, \frac{1}{x} \, dx \]

Step 5: Conclude the Solution

Notice that the integral \( \int \ln x \, \frac{1}{x} \, dx \) repeats itself. Therefore, solve it by recognizing this is a differential equation or an integral leading back to \( (\ln x)^2 \). Thus the result doesn't further simplify with elementary functions beyond the substitution made.Conclude with:\[ \int \frac{\ln x^2}{x} \, dx = (\ln x)^2 + C \] where \( C \) is the constant of integration.

Key Concepts

Integration by Parts

Integration by parts is a helpful technique derived from the product rule for differentiation. It's particularly useful when trying to integrate the product of functions that do not easily fit into basic integration rules. To apply integration by parts, we use the formula:\[\int u \, dv = uv - \int v \, du\]Here, you start by choosing which part of your integrand will be \( u \) and which part will be \( dv \). A strategic choice simplifies the integral you will address, often simplifying the work at hand by reducing complex functions step by step. For example, in problems involving logarithmic functions, picking \( u = \ln x \) and \( dv = \frac{1}{x} \, dx \) works well, as this choice simplifies the process when integrating and differentiating these components. Remember, if your integral simplifies back to a repeated form, often you need to solve out once more with a simpler substitution or recognize that it cycles, providing a hint at the final solution.

Logarithmic Functions

Logarithmic functions, like \( \ln x \), arise frequently in integration problems due to their unique properties and relationships with exponential functions. Understanding logarithmic properties, such as \( \ln x^2 = 2 \ln x \), allows you to rewrite and simplify complex expressions before tackling the integration process.When integrating functions that include logarithmic components, the choice of substitution matters greatly. For instance, transforming the integrand by using logarithmic properties can turn a seemingly difficult integral into one with a straightforward path using methods like substitution or integration by parts.For this process, it's crucial to remember common logarithm rules:

• \( \ln(xy) = \ln x + \ln y \)
• \( \ln \left(\frac{x}{y}\right) = \ln x - \ln y \)
• \( \ln(x^n) = n \ln x \)

These properties often provide the clue needed to simplify your integral, and without rewriting expressions cannot lead to the simplest form before diving into integration.

Definite and Indefinite Integrals

The primary difference between definite and indefinite integrals lies in their outcomes and use cases. An indefinite integral, represented by \( \int f(x) \, dx \), provides a function's antiderivative plus a constant \( C \), as this integral represents a family of solutions. The addition of \( C \) accounts for any constant shift along the \( y \)-axis the original function might have had.In contrast, a definite integral is associated with limits of integration and represents the net area under a curve from one point to another, providing a specific numerical result. This integral takes the form:\[\int_{a}^{b} f(x) \, dx\]where \( a \) and \( b \) are the limits of integration.With indefinite integrals, like in the exercise provided, it is purposed not with finding an explicit solution but rather understanding a broader family of solutions. So every time integration by parts, substitution, or straightforward calculation modifies \( f(x) \), the ultimate goal is to wrap the result neatly into an expression with constant \( C \) included as needed.