/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 70 Evaluate the integrals. \(\int... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 70

Evaluate the integrals. \(\int \frac{d x}{x\left(\log _{8} x\right)^{2}}\)

Short Answer

Expert verified
The integral is \(-\frac{(\ln 8)^2}{\ln x} + C\)."

Step by step solution

01

Variable Substitution

To simplify the integral, use a substitution. Let \( u = \log_{8} x \). Then, we have \( x = 8^u \) and \( dx = 8^u \ln(8) \, du \). Substituting these into the integral yields:\[\int \frac{dx}{x (\log_{8} x)^2} = \int \frac{8^u \ln(8) \, du}{8^u u^2} = \int \frac{\ln(8) \, du}{u^2}\]
02

Integrate Using Power Rule

Now, integrate \( \int \frac{\ln(8)}{u^2} \, du \). We can factor out the constant \( \ln(8) \), giving:\[\ln(8) \int u^{-2} \, du\]The integral of \( u^{-2} \) is \( -u^{-1} \) (power rule for integration). Therefore:\[\ln(8) \left( -\frac{1}{u} \right) + C = -\frac{\ln(8)}{u} + C\]
03

Substitute Back

Replace \( u \) with \( \log_{8} x \) to express the integral in terms of \( x \):\[-\frac{\ln(8)}{\log_{8} x} + C\]
04

Simplify Using Change of Base Formula

Use the change of base formula \( \log_{8} x = \frac{\ln x}{\ln 8} \) to further simplify the expression:\[-\frac{\ln(8)}{\frac{\ln x}{\ln 8}} = -\frac{(\ln(8))^2}{\ln x} + C\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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 complex integrals by transforming them into simpler ones. In this exercise, we used the substitution \( u = \log_{8} x \). This transformation simplifies the integrative process:
  • First, express \( x \) in terms of \( u \), giving \( x = 8^u \).
  • Calculate \( dx \) in terms of \( du \) to replace it in the integral, so \( dx = 8^u \ln(8) \, du \).
  • Substitute \( x \) and \( dx \) in the original integral. This simplifies the complex expression into a simpler form: \( \int \frac{\ln(8) \, du}{u^2} \).
Substitution transforms the integral into a more manageable form, where standard techniques like the power rule can be applied. This is crucial for calculating integrals that may initially seem difficult or impossible.
Integral Calculus
Integral calculus is the process of finding the integral or antiderivative of functions. It includes a variety of techniques, such as the substitution method, which we used in this exercise.
  • An integral essentially calculates the area under a curve. In mathematical terms, it is the reverse of differentiation.
  • Integrals can be indefinite, as in this exercise, which includes a constant \( C \) to account for any constants lost during differentiation.
  • Using the power rule for integration helps find antiderivatives for functions of form \( u^n \).
In this example, the integral of \( u^{-2} \) was evaluated using the power rule: the antiderivative is \( -u^{-1} \). This results in an expression that's easier to interpret or to be transformed back into the original variable.
Logarithmic Integration
Logarithmic integration comes into play when the integrand involves logarithmic expressions, such as \( \log_{8} x \) in this exercise. We see:
  • Substitutions often use logarithmic forms to transform and simplify. Here, \( \log_{8} x \) becomes \( u \), which streamlines the integration process.
  • Understanding properties such as the change of base formula \( \log_{8} x = \frac{\ln x}{\ln 8} \) is essential for simplifying and interpreting integral results.
After substituting back into terms of \( x \), we utilized this change of base formula. This reduced the expression to a simpler form. It demonstrated the elegance of logarithmic integration, making seemingly complex problems more approachable.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate the integrals in Exercises \(67-74\) in terms of a. inverse hyperbolic functions. b. natural logarithms. $$ \int_{0}^{2 \sqrt{3}} \frac{d x}{\sqrt{4+x^{2}}} $$

The region between the curve \(y=\sec ^{-1} x\) and the \(x\) -axis from \(x=1\) to \(x=2\) (shown here) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid.

Evaluate the integrals in Exercises \(71-94\) $$ \int \frac{d x}{(2 x-1) \sqrt{(2 x-1)^{2}-4}} $$

Evaluate the integrals in Exercises \(51-60 .\) $$ \int_{1}^{2} \frac{\cosh (\ln t)}{t} d t $$

Can the integrations in (a) and (b) both be correct? Explain. $$ \begin{array}{l}{\text { a. } \int \frac{d x}{\sqrt{1-x^{2}}}=\sin ^{-1} x+C} \\\ {\text { b. } \int \frac{d x}{\sqrt{1-x^{2}}}=-\int-\frac{d x}{\sqrt{1-x^{2}}}=-\cos ^{-1} x+C}\end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.