Problem 51 - AP Calculus with 8 Practice Tests 15th

Question

\(\frac{\ln \sqrt{x}}{x} d x=\) (A) \(\frac{\ln ^{2} \sqrt{x}}{\sqrt{x}}+C\) (B) \(\frac{1}{2} \ln |\ln x|+C\) (C) \(\frac{(\ln \sqrt{x})^{2}}{2}+C\) (D) \(\frac{1}{4} \ln ^{2} x+C\)

Step-by-Step Solution

Answer

The correct answer is (D) \(\frac{1}{4} \ln^2 x + C\).

Step 1: Simplify the Expression Inside the Integral

We begin by simplifying the term inside the integral, \( \frac{\ln \sqrt{x}}{x} \).Recall that \( \ln \sqrt{x} = \ln x^{1/2} = \frac{1}{2} \ln x \), so the expression becomes \( \frac{\frac{1}{2} \ln x}{x} = \frac{1}{2} \cdot \frac{\ln x}{x} \).\Thus, we need to integrate \( \frac{1}{2} \cdot \frac{\ln x}{x} \) with respect to \( x \).

Step 2: Identify the Type of Integral

The integral \( \int \frac{\ln x}{x} \, dx \) is a standard integral that can be solved using integration by parts.Integration by parts formula is \( \int u \, dv = uv - \int v \, du \). Here, we can let \( u = \ln x \) and \( dv = \frac{1}{x} \, dx \).

Step 3: Apply Integration by Parts

With \( u = \ln x \) and \( dv = \frac{1}{x} \, dx \), compute \( du = \frac{1}{x} \, dx \) and \( v = \int \frac{1}{x} \, dx = \ln x \).Applying the integration by parts formula, \( \int \ln x \cdot \frac{1}{x} \, dx = \ln x \cdot \ln x - \int \ln x \cdot \frac{1}{x} \, dx = \frac{1}{2} (\ln x)^2 \).

Step 4: Multiply by Constant

Since the integral we are solving is \( \frac{1}{2} \cdot \int \frac{\ln x}{x} \, dx \), we multiply the result of the integral by \( \frac{1}{2} \).This gives us \( \frac{1}{2} \cdot \frac{1}{2} (\ln x)^2 = \frac{1}{4} (\ln x)^2 \cdot C \).

Step 5: Verify with the Given Options

Checking the expression \( \frac{1}{4} (\ln x)^2 + C \) against the options, we find that it corresponds to option (D).

Key Concepts

Logarithmic Integration

Logarithmic integration involves integrating expressions that contain logarithmic functions, such as \( \ln x \). This often requires the use of specific techniques because logarithms behave differently from algebraic functions. When dealing with integrals of the form \( \frac{\ln x}{x} \), simplifying the expression is a helpful first step.
A common approach when handling logarithmic integrals is to simplify the logarithmic term using properties of logarithms. This allows for easier manipulation within the integral setup. In our example, \( \ln \sqrt{x} \) was transformed into \( \frac{1}{2} \ln x \) to simplify the integration process.
Often, integration by parts is used to solve logarithmic integrals because it provides a structured way to transform the integral into a more workable form. By recognizing the logarithmic part as \( u \) and then identifying \( dv \) appropriately, we can leverage this technique to solve the integral effectively.

Integral Calculus

Integral calculus is the branch of mathematics concerning the accumulation of quantities, as well as the areas underneath and between curves. It is a powerful tool used in various fields like physics, engineering, and economics.
The core purpose of integral calculus is to find the integral, or antiderivative, of a function, giving a new function whose derivative is the original function. In our exercise, the focus was on integrating a function involving a logarithm, which requires the practitioner to apply specific integration techniques like substitution or integration by parts.

• Methods such as substitution are often used when integrands have composite functions.
• Integration by parts is useful when dealing with products of functions, such as those involving logarithmic and algebraic expressions.

Careful analysis and step-by-step techniques, as demonstrated in solving this exercise, help simplify complex integrals into more recognizable forms leading to the solution.

Calculus Problem Solving

Calculus problem solving involves a strategic approach to understanding and breaking down complex mathematical problems. The process typically includes:

• Identifying the type of problem: Which calculus technique can best be applied?
• Breaking down the problem into simpler, more manageable parts.
• Applying relevant calculus techniques step-by-step.

In this exercise, the problem required recognizing how the logarithmic function could be integrated through simplification and knowledge of integration by parts. Calculus problems often present themselves with various options, requiring this methodical approach.
Each step in problem-solving helps build towards the final goal. Simplifying the expression \( \frac{\ln \sqrt{x}}{x} \), identifying the integral type, and selecting integration by parts necessary for further simplification are all part of the strategy. This ensures the correct application of calculus techniques to solve a complex problem effectively and accurately.