Problem 44 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int_{2}^{4} \frac{d x}{x \ln x}$$

Step-by-Step Solution

Answer

The integral evaluates to \( \ln\left(\frac{\ln 4}{\ln 2}\right) \).

Step 1: Recognizing the Type of Integral

This integral requires a method that handles the structure \( \frac{1}{x \ln x} \). Notice that the integral does not look like a standard textbook example directly. We need to recognize a way to transform it for easier integration.

Step 2: Substitution Method

Use substitution to simplify the integral. Let \( u = \ln x \), which implies that \( du = \frac{1}{x} dx \). This makes \( dx = x du \), or rewritten using \( x = e^u \) in the substitution, we get \( dx = e^u du \). The integral bounds transform as follows: when \( x = 2 \), \( u = \ln 2 \); and when \( x = 4 \), \( u = \ln 4 \).

Step 3: Simplifying the Integral

With the substitution \( u = \ln x \) and \( dx = e^u du \), the integral \( \int \frac{d x}{x \ln x} \) transforms into \( \int_{\ln 2}^{\ln 4} \frac{e^u du}{e^u u} \). This cancels the \( e^u \), simplifying the integral to \( \int_{\ln 2}^{\ln 4} \frac{du}{u} \).

Step 4: Solving the Integral

The integral \( \int \frac{du}{u} \) is a standard natural logarithm integral, which evaluates to \( \ln |u| + C \). Since this is a definite integral, we have: \[ \ln |u| \bigg|_{\ln 2}^{\ln 4} = \ln(\ln 4) - \ln(\ln 2). \]

Step 5: Calculating the Result

Compute the values by plugging in the limits to get the difference. So the result of the integral is: \( \ln(\ln 4) - \ln(\ln 2) = \ln\left(\frac{\ln 4}{\ln 2}\right) \).

Key Concepts

Definite Integrals

Definite integrals are used to find the exact area under a curve between two specific points. They are written with limits on the integral sign. For example, \( \int_{a}^{b} f(x) \, dx \) is a definite integral where \( a \) and \( b \) are the lower and upper limits respectively.

• The function \( f(x) \) is integrated between these limits to find the total area.
• The calculation gives a numerical value instead of another function.

Understanding definite integrals is crucial in real-world applications. They help in

• calculating areas under curves,
• computing probabilities in statistics, and
• solving physics problems involving motion.

In the exercise, the definite integral \( \int_{2}^{4} \frac{dx}{x \ln x} \) requires finding the area under the curve \( \frac{1}{x \ln x} \) from \( x = 2 \) to \( x = 4 \). This integral isn’t straightforward due to the complex function, which leads us to use a different method called substitution.

Substitution Method

The substitution method is essential when dealing with complex integrals. It simplifies integrals by transforming the variable, reducing the integral into a more standard form. Here's how substitution works:

• Choose a substitution that simplifies the integral. In this exercise, we use \( u = \ln x \).
• Find the derivative, \( du = \frac{1}{x} dx \), which allows replacing \( dx \) in the integral.
• Change the bounds according to the substitution. When \( x = 2 \), \( u = \ln 2 \); when \( x = 4 \), \( u = \ln 4 \).

With these transformations, the integral becomes \( \int_{\ln 2}^{\ln 4} \frac{du}{u} \). This simplifies calculations, making it easier to find the solution for the definite integral.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is the logarithm with base \( e \) (approximately 2.718). It's a fundamental concept in calculus, often appearing in integration and differentiation.

1. \( \ln x \) indicates the power to which \( e \) must be raised to obtain \( x \).
2. It is especially useful in continuous growth models such as in finance and biology.

In the context of integration, the antiderivative of \( \frac{1}{x} \) is \( \ln |x| + C \). This formula is crucial when solving integrals like \( \int \frac{1}{u} \, du = \ln |u| + C \). For our specific exercise, after using substitution, the integral simplifies to \( \int_{\ln 2}^{\ln 4} \frac{du}{u} \), which results in \( \ln(\ln 4) - \ln(\ln 2) \). This shows the usefulness of natural logarithms in converting complex expressions into more manageable forms, allowing for easy evaluation of definite integrals.