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Chapter 7: Problem 12

Evaluate the integrals. $$\int \frac{\ln (\ln x)}{x \ln x} d x$$

Short Answer

Expert verified
\( \int \frac{\ln (\ln x)}{x \ln x} \, dx = \frac{(\ln (\ln x))^2}{2} + C \).

Step by step solution

01

Identify the substitution

Observe that the given integral is \( \int \frac{\ln (\ln x)}{x \ln x} \, dx \). Consider a substitution that simplifies \( \ln x \). Let \( u = \ln x \). Then, \( du = \frac{1}{x} \, dx \).
02

Perform the substitution

Substitute \( u = \ln x \) and \( du = \frac{1}{x} \, dx \) into the integral. This changes the integral to \( \int \frac{\ln u}{u} \, du \).
03

Integrate the new function

The integral \( \int \frac{\ln u}{u} \, du \) is a standard form. Recognize that this integral can be solved as \( \frac{(\ln u)^2}{2} + C \), where \( C \) is the constant of integration.
04

Substitute back to the original variable

Replace \( u \) with \( \ln x \) to revert to the original variable. The solution becomes \( \frac{(\ln (\ln x))^2}{2} + C \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a powerful tool in integral calculus that simplifies integration by changing variables. This technique is similar to the chain rule used in differentiation. The goal is to transform a complex integral into a simpler one, which might be more straightforward to solve.
Consider the integral \( \int \frac{\ln (\ln x)}{x \ln x} \, dx \). It may initially seem challenging, but by using a suitable substitution, it becomes manageable.
  • Choose a substitution: Select a part of the integrand that can be replaced to simplify the expression. In this case, let \( u = \ln x \).
  • Determine \( du \): Differentiate \( u \) with respect to \( x \), giving \( du = \frac{1}{x} \, dx \).
  • Substitute: Replace \( \ln x \) with \( u \) and \( dx \) with \( x \, du \), transforming the integral into \( \int \frac{\ln u}{u} \, du \).
This new integral is easier to handle, allowing for a more straightforward evaluation.
Natural Logarithm
The natural logarithm, denoted as \( \ln(x) \), is a fundamental mathematical function. It is specifically the logarithm to the base \( e \), where \( e \) is approximately 2.71828. The natural logarithm has several unique properties that make it useful in calculus and mathematical modeling.
  • Inverse of the exponential function: \( \ln(x) \) is the inverse function of \( e^x \).
  • Derivative: The derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \).
  • Integral: The integral of \( \frac{1}{x} \) is \( \ln|x| + C \), where \( C \) is the constant of integration.
When integrating expressions involving natural logarithms, it is often helpful to use substitution, as seen in the exercise. Here, we use \( \ln x \) in our substitution to break down the complexity of the original function.
Definite Integrals
Definite integrals extend the concept of an indefinite integral, specifically calculating the area under a curve within defined limits. While the original exercise dealt with an indefinite integral, understanding definite integrals provides further insight into the application of integration techniques.
  • Limits of integration: A definite integral has specified upper and lower bounds, signified by numbers placed at the top and bottom of the integral sign, respectively.
  • Net area: Unlike indefinite integrals, definite integrals yield a numerical result, representing the net area between the curve and the \( x \)-axis over the interval.
  • Fundamental Theorem of Calculus: This theorem connects differentiation and integration, stating that if a function is continuous on a closed interval \([a, b]\), the integral of its derivative over [a, b] is equal to the change in its original function.
Knowing how to evaluate definite integrals is crucial for applications in physics, engineering, and other fields where calculating total quantities, like work or area, is necessary. It's important to understand definite integrals to put substitution and other integration techniques into broader use.

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Most popular questions from this chapter

Evaluate the integrals. $$\int_{0}^{\ln 10} 4 \sinh ^{2}\left(\frac{x}{2}\right) d x$$

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