/*! 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 16 Write the general antiderivative... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 16

Write the general antiderivative. \(\int \frac{1}{x(\ln x)^{2}} d x\)

Short Answer

Expert verified
The general antiderivative is \( -\frac{1}{\ln x} + C \).

Step by step solution

01

Identify the Integral Type

The integral given is \( \int \frac{1}{x(\ln x)^{2}} \, dx \), which appears to be a form that could be solved using substitution, as it involves a composite function \( \ln x \).
02

Choose a Suitable Substitution

Let's set \( u = \ln x \). Then \( du = \frac{1}{x} dx \). This substitution simplifies the integral into a simpler form since the \( \frac{1}{x} \) in the integral corresponds to \( du \).
03

Rewrite the Integral

Substitute \( u = \ln x \) and \( du = \frac{1}{x} dx \) into the integral. The integral becomes \( \int \frac{1}{u^2} \, du \).
04

Integrate the Simplified Expression

The integral \( \int \frac{1}{u^2} \, du \) can be rewritten as \( \int u^{-2} \, du \). The antiderivative of \( u^{-n} \) is \( \frac{u^{-n+1}}{-n+1} + C \) when \( n eq 1 \). Thus, integrating, we have \( -u^{-1} + C \).
05

Substitute Back to the Original Variable

Substitute \( u = \ln x \) back into the antiderivative to obtain the result in terms of \( x \). \(-\frac{1}{u} + C = -\frac{1}{\ln x} + C \).
06

Write the General Antiderivative

The general antiderivative of \( \int \frac{1}{x(\ln x)^2} \, dx \) is \( -\frac{1}{\ln x} + C \), where \( C \) is the constant of integration.

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.

Antiderivative
In integral calculus, the antiderivative is a fundamental concept. It refers to finding a function whose derivative matches a given function. The process of finding an antiderivative is also known as integration.
The notation for an antiderivative involves the integral sign, followed by the function and the differential variable, such as \( \int f(x) \, dx \). The antiderivative is not unique, as it includes an arbitrary constant denoted by \( C \,\), known as the constant of integration.
  • An antiderivative of a function \( f(x) \) satisfies \( F'(x) = f(x) \).
  • The general solution is \( F(x) + C \), where \( C \) can be any constant.
Finding an antiderivative is crucial for solving differential equations and for calculating the total change from rates.
Substitution Method
The substitution method, sometimes known as "\( u \)-substitution," is a technique used to make integration easier. It is particularly useful when dealing with composite functions, as it simplifies the integral into a form that is easier to manage. The choice of substitution is key; it often involves selecting a function within the integral whose derivative is present elsewhere in the integrand.
  • Begin by choosing \( u = g(x) \), where \( g(x) \) is a part of the integrand.
  • Find the differential \( du = g'(x) \, dx \).
  • Rewrite the integral in terms of \( u \).
  • Once integrated, substitute back the original function for \( u \).
This method simplifies the calculation by transforming difficult integrals into basic ones that are straightforward to integrate.
Natural Logarithm
The natural logarithm, denoted as \( \ln x \), is a logarithmic function with the base \( e \), where \( e \approx 2.718 \). It is widely used in calculus due to its unique properties, especially regarding differentiation and integration.
An essential property is that its derivative is \( \frac{1}{x} \,\) making it pivotal in many integration problems, such as in the given exercise.
  • The integral \( \int \frac{1}{x} \, dx = \ln |x| + C \).
  • Natural logarithms aid in solving exponential growth models due to their relationship with the exponential function \( e^x \).
The presence of \( \ln x \) as a part of a composite function often indicates that substitution could simplify the integral.
Composite Function
A composite function is formed when one function becomes the input of another, expressed as \( f(g(x)) \). In integration, dealing with composite functions can be tricky without the right techniques.
The substitution method is often employed to simplify the integration process by identifying a substitution that can handle the inner function.
  • Identify \( g(x) \) and its derivative \( g'(x) \) within the integral.
  • Select \( u = g(x) \).
  • Transform the integral into a simpler form using \( u \).
Composite functions are common in calculus as they represent a broad range of real-world phenomena and require methods like the substitution method for effective integration.

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

a. Calculate the total area of the region(s) between the graph of \(f\) and the \(x\) -axis from \(a\) to \(b\). b. Evaluate \(\int_{a}^{b} f(x) d x\) c. Explain why the result from part \(a\) differs from that of part \(b\). $$ f(x)=\frac{9.295}{x}-1.472 ; a=5, b=10 $$

Foreign Trade The rate of change of the value of goods exported from the United States between 1990 and 2001 can be modeled as \(E^{\prime}(t)=-1.665 t^{2}+16.475 t+7.632\) billion dollars per year and the rate of change of the value of goods imported into the United States during those years can be modeled as \(I^{\prime}(t)=4.912 t+40.861\) billion dollars per year where \(t\) is the number of years since \(1990 .\) a. Calculate the difference between the accumulated value of imports and the accumulated value of exports from the end of 1990 through 2001 b. Is the answer from part \(a\) the same as the area of the region(s) between the graphs of \(E^{\prime}\) and \(I^{\prime} ?\) Explain.

a. Write the formula for \(\int f(x) d x\). b. Write the formula for \(\frac{d}{d x} \int f(x) d x\). $$ f(x)=\frac{25}{x^{4}} $$

Oil Production \(\quad\) On the basis of data obtained from a preliminary report by a geological survey team, it is estimated that for the first 10 years of production, a certain oil well in Texas can be expected to produce oil at the rate of \(r(t)=3.935 t^{3.55} e^{-1.351 t}\) thousand barrels per year where \(t\) is the number of years after production begins. a. Calculate the average annual yield from this oil well during the first 10 years of production. b. During the first 10 years, when is the rate of yield expected to be equal to the 10 -year average annual yield?

a. write the general antiderivative, b. evaluate the expression. \(\int_{0}^{2} e^{\sin x} \cos x d x\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.