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Chapter 1: Problem 331

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

Short Answer

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

Step by step solution

01

Identify the Substitution

We need to find an appropriate substitution for the integral \( \int \frac{dx}{x \ln x \ln (\ln x)} \). Notice that we have multiple nested logarithms in the denominator. A helpful substitution can involve breaking down these logs. Let's set \( u = \ln (\ln x) \). Then, the derivative of \( u \) with respect to \( x \) will help simplify the integral.
02

Compute the Derivative for Substitution

From the chosen substitution \( u = \ln (\ln x) \), we find the derivative using the chain rule: Since \( \frac{d}{dx} (\ln x) = \frac{1}{x} \), applying the chain rule again for \( \ln (\ln x) \), we have: \( \frac{du}{dx} = \frac{1}{\ln x} \cdot \frac{1}{x} = \frac{1}{x \ln x} \). This simplifies substitution later.
03

Rewrite the Integral Using Substitution

We rearrange the derivative to express \( dx \): Using \( \frac{du}{dx} = \frac{1}{x \ln x} \), rearrange to get \( dx = x \ln x \cdot du \). Substitute this into the original integral: \[ \int \frac{dx}{x \ln x \ln (\ln x)} = \int \frac{x \ln x \cdot du}{x \ln x \cdot u} = \int \frac{1}{u} \, du \]. This simplifies the integral significantly.
04

Solve the Simplified Integral

The integral \( \int \frac{1}{u} \, du \) is a standard integral that equals \( \ln |u| + C \), where \( C \) is the constant of integration. Thus, \[ \int \frac{dx}{x \ln x \ln (\ln x)} = \ln |u| + C \].
05

Back-Substitute to the Original Variable

Using our substitution \( u = \ln (\ln x) \), substitute back to express the solution in terms of \( x \): \( \ln |u| = \ln |\ln (\ln x)| \). Thus, the integral \( \int \frac{dx}{x \ln x \ln (\ln x)} = \ln |\ln (\ln x)| + C \). This is the final result.

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Key Concepts

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

Integration by Substitution
Integration by substitution is like finding a detour that makes the journey easier. It's a technique used to simplify complex integrals. The idea is to transform a difficult integral into a simpler one by making an appropriate substitution for a variable.
  • Think of it as changing the variable from one letter to another, making the equation more manageable.
  • In our example, the goal was to simplify the expression involving multiple logarithms.
For the given integral, the substitution chosen is \( u = \ln(\ln x) \). This substitution helps to simplify complex parts of the expression, making it easier to integrate. Once you've chosen a substitution, the next step involves finding the derivative and rewriting the integral in terms of the new variable.
Nested Logarithms
Nested logarithms involve having one logarithmic function inside another. This creates a layer of complexity because you are essentially "logging" a log.
  • An example of a nested logarithm is \( \ln(\ln x) \).
  • This kind of expression can lead to confusion and requires careful handling, often through substitution.
In our task, the nested log \( \ln(\ln x) \) played a crucial role. By setting \( u = \ln(\ln x) \), we unfold these layers, making the integral simpler and more manageable. This strategic approach helps deal with such complex expressions effectively.
Chain Rule
The chain rule is a critical differentiation tool, especially when dealing with complex functions. It's used to find the derivative of a composition of functions 鈥� one function inside another.
  • It's like peeling an onion layer by layer when you need the derivative.
  • In our case, it helped differentiate \( \ln(\ln x) \).
To apply the chain rule here, consider \( u = \ln(\ln x) \). We first differentiate \( \ln x \) to get \( \frac{1}{x} \), then differentiate \( \ln(\ln x) \) using the chain rule, resulting in \( \frac{1}{x \ln x} \). This derivative is essential for rewriting the original variable in terms of \( u \).
Constant of Integration
When finding indefinite integrals, you must include a constant of integration. This constant, usually denoted as \( C \), represents the infinite set of all antiderivatives.
  • It accounts for any vertical shift in the function's graph.
  • Without \( C \), the general solution to the integral wouldn't be complete.
In the context of our solution, after integrating \( \frac{1}{u} \) to get \( \ln |u| \), we add \( C \) to denote the complete family of solutions. This constant reminds us that there are many possible solutions that differ by a constant. Including \( C \) is crucial to indicating all possible antiderivatives of the function being integrated.

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Most popular questions from this chapter

In the following exercises, use a change of variables to show that each definite integral is equal to zero. $$ \int_{\pi / 4}^{3 \pi / 4} \sin ^{2} t \cos t d t $$

Show that the average value of \(\sin ^{2} t\) over \([0,2 \pi]\) is equal to \(1 / 2\) Without further calculation, determine whether the average value of \(\sin ^{2} t\) over \([0, \pi]\) is also equal to \(1 / 2\).

Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln x,\) but keep in mind that \(\ln x\) has an inverse function defined on \((-\infty, \infty) .\) Call it \(E\). Show that \(E^{\prime}(t)=E(t)\).

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \frac{x^{2}}{\left(x^{3}-3\right)^{2}} d x $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int\left(\sin ^{2} \theta-2 \sin \theta\right)\left(\sin ^{3} \theta-3 \sin ^{2} \theta\right)^{3} \cos \theta d \theta $$
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