Problem 44 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(37-54\). $$ \int_{2}^{4} \frac{d x}{x \ln x} $$

Step-by-Step Solution

Answer

The value of the integral is \( \ln 2 \).

Step 1: Identify the Integral Form

The given integral requires evaluating \( \int_{2}^{4} \frac{dx}{x \ln x} \). This integrand is of the form \( \frac{1}{x \ln x} \), which suggests that a substitution might simplify the integration process.

Step 2: Choose a Substitution

To solve this integral, we will use the substitution \( u = \ln x \). The differential \( du \) will then be \( du = \frac{1}{x} dx \), or equivalently, \( dx = x du \). This substitution will simplify our integrand.

Step 3: Change the Limits of Integration

When \( x = 2 \), \( u = \ln 2 \). When \( x = 4 \), \( u = \ln 4 \). So, after substitution, the limits of integration change from \([2, 4]\) to \([\ln 2, \ln 4]\).

Step 4: Substitute and Simplify

Substitute \( u = \ln x \) and \( dx = x du = e^u du \) in the integral. The integral becomes \( \int_{\ln 2}^{\ln 4} \frac{e^u du}{e^u u} = \int_{\ln 2}^{\ln 4} \frac{du}{u} \).

Step 5: Evaluate the Integral

The integral \( \int \frac{du}{u} \) is a standard result, which gives \( \ln |u| + C \). Applying the limits of integration: \[ \left[ \ln |u| \right]_{\ln 2}^{\ln 4} = \ln (\ln 4) - \ln (\ln 2) \].

Step 6: Simplify the Result

Using properties of logarithms, \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \), simplify the expression: \[ \ln\left( \frac{\ln 4}{\ln 2} \right) = \ln(2) \].

Key Concepts

Integration by Substitution

Integration by substitution, sometimes called "u-substitution," is a technique often used to make integrals easier to evaluate. The idea is to transform the integral into a simpler form by substituting part of the integrand with a new variable. This is particularly useful when dealing with composite functions or when an expression suggests that a derivative appears within the integrand.

To use integration by substitution, start by identifying a substitution that aligns a part of the integral with its differential. In our specific example of integrating \( \frac{1}{x \ln x} \), we noticed an opportunity to simplify the problem by setting \( u = \ln x \).

• Choosing \( u = \ln x \) made sense because the derivative \( du = \frac{1}{x} dx \) is already present.
• This transforms the integral into one of the natural logarithmic form, easy to integrate.
• After substituting, always remember to change the limits of integration accordingly, converting them from \( x \) values to \( u \) values.

Overall, substitution turned a complex problem into a much simpler one, highlighting its power in calculus integrations.

Definite Integrals

Definite integrals differ from indefinite integrals by the presence of limits, which signify the range over which the function is evaluated. In this exercise, those limits changed due to the substitution. The substitution process requires adjusting limits to fit the new variable.

In our calculation, we initially had limits from \( x = 2 \) to \( x = 4 \). With the substitution, these limits converted into \( u = \ln 2 \) and \( u = \ln 4 \).

• Definite integrals provide a numeric result which represents the area under the curve between the limits.
• After applying a substitution, it is crucial to reevaluate the limits according to the new variable.

This characteristic differentiates definite from indefinite integrals, whose results include a constant. Remember, there's no constant \( C \) included in definite integrals as they are calculated between two bounds.

Logarithmic Integration

Logarithmic integration involves integrating functions that include a natural logarithm. It is a frequently encountered scenario in calculus, especially in functions similar to the one in our problem: \( \int \frac{1}{x \ln x} \).

Upon utilizing the substitution \( u = \ln x \), the integral transformed into \( \int \frac{du}{u} \), which is a classic form encountered in calculus. This integral is straightforward because the antiderivative of \( \frac{1}{u} \) is \( \ln |u| \).

• The logarithmic form is easy to integrate due to these standard antiderivatives.
• Once integrated, apply the limits to find the numeric solution, enhancing the skill of recognizing logarithmic forms.
• Simplifying the result involves using properties of logarithms, such as \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \).

This problem shows how logarithmic integration can take a complex integral and transform it into a manageable one with straightforward results.