Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$ \int \frac{d x}{x \ln x \ln (\ln x)} $$

Short answer

The indefinite integral is \( \ln |\ln(\ln x)| + C \).

Step-by-step solution

Identify the Inner Functions for Substitution

The integrand, \( \frac{1}{x \ln x \ln ( \ln x)} \), suggests that substitution might simplify the integration. First, identify possible candidates for substitution. The innermost function among \( x, \ln x, \ln(\ln x) \) is \( \ln(\ln x) \). Thus, choose \( u = \ln(\ln x) \), which suggests \( du = \frac{1}{\ln x}\cdot\frac{1}{x}dx \). Observe that \( du \) is a match for part of the differential in our integrand, \( \frac{1}{x \ln x}dx \).

Perform the Substitution

Substituting \( u = \ln(\ln x) \), we have \( du = \frac{1}{x \ln x} \, dx \). Therefore, the integral transforms as follows:\[ \int \frac{1}{x \ln x \ln ( \ln x)} \, dx = \int \frac{1}{u} \, du \].

Integrate the Simpler Expression

After substitution, the integral becomes \( \int \frac{1}{u} \, du \), which is a basic integral form. The integral of \( \frac{1}{u} \) with respect to \( u \) is \( \ln |u| + C \), where \( C \) is the constant of integration.

Back Substitution

Replace \( u \) with the original expression in terms of \( x \). Since we set \( u = \ln(\ln x) \), substitute back to obtain:\[ \ln |\ln(\ln x)| + C \].

Write the Final Answer

The indefinite integral of the given function \( \int \frac{dx}{x \ln x \ln (\ln x)} \) evaluates to:\[ \ln |\ln(\ln x)| + C \], where \( C \) is the constant of integration.

Key concepts

Substitution Method

The substitution method is a powerful tool for solving integrals. It helps simplify complex integrals by replacing parts of the integral with a new variable. This is particularly useful when integrating nested functions or complex expressions. For example, in the integral \( \int \frac{1}{x \ln x \ln(\ln x)} \, dx \), we can identify the innermost part as \( \ln(\ln x) \). By letting \( u = \ln(\ln x) \), we transform the integration variable to \( du \). ### Steps in Substitution- **Identify the Inner Function:** Choose an inner part of the integrand. This could be the most nested function, as it often simplifies expressions effectively. - **Find the Differential:** Derive \( du \) from \( u \). This involves differentiating \( u \) with respect to \( x \) and matching it to a part of the integrand.- **Simplify and Integrate:** Use the new variable to transform the integral into a more manageable form.- **Back Substitute:** Once integrated, replace \( u \) with the original expression in terms of \( x \). The goal is to transition the original complex integral to a simpler form, making the problem more approachable.

Logarithmic Integration

Logarithmic integration is a technique where you integrate expressions involving logarithmic functions. Often, substitution will transform the original integral into a logarithmic form. In our example, after substitution, the integral became \( \int \frac{1}{u} \, du \). This is a classic case for logarithmic integration. The result of integrating \( \frac{1}{u} \) is \( \ln|u| + C \), where \( C \) is the integration constant. ### Why Logarithmic Integration?- **Direct Evaluation:** Logarithmic integrals are straightforward because the integral of \( \frac{1}{u} \) directly gives a logarithmic function.- **Common Appearances:** Many problems involving exponential growth or decay present integrands like \( \frac{1}{x} \), making this a frequently used technique. By applying the substitution method and recognizing logarithmic forms, we can efficiently integrate otherwise daunting expressions.

Basic Integral Forms

Basic integral forms are essential building blocks for understanding and solving integrals. They are the fundamental integrals that you often come across, such as \( \int x^n \, dx \) or \( \int e^x \, dx \).In the given exercise, identifying the basic form \( \int \frac{1}{u} \, du \) is crucial. This form links directly to logarithmic integration. Recognizing these forms helps to quickly apply integral rules and find solutions.### Common Basic Form Examples

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) (\( n eq -1 \))
• \( \int \frac{1}{x} \, dx = \ln |x| + C \)
• \( \int e^x \, dx = e^x + C \)

Understanding and memorizing these forms can dramatically speed up the integration process. They form the foundation for more complex techniques, like those used in substitution and logarithmic integration, making them indispensable for calculus students.