Problem 69
Question
Evaluate the integrals. \(\int \frac{d x}{x \log _{10} x}\)
Step-by-Step Solution
Verified Answer
\( \ln(10) \ln |\log_{10} x| + C \).
1Step 1: Identify the Type of Integral
The given integral is \( \int \frac{dx}{x \log_{10} x} \). This is a type of integral that requires substitution due to the combination of functions \(x\) and \(\log_{10} x\) in the denominator.
2Step 2: Choose a Suitable Substitution
To simplify the integral, use substitution. Let \( u = \log_{10} x \). Then the derivative \( \frac{du}{dx} \) becomes \( \frac{1}{x}\ln(10) \), thus \( du = \frac{1}{x \ln(10)} dx \). Rearrange this to solve for \( dx \): \( dx = x \ln(10) du \).
3Step 3: Substitute and Simplify
Substitute \( u \) and \( dx \) into the integral: \[ \int \frac{dx}{x \log_{10} x} = \int \frac{x \ln(10) du}{x u} = \int \frac{\ln(10)}{u} du. \] The \( x \) terms cancel out, leaving a simpler integral to evaluate.
4Step 4: Integrate the Simplified Expression
This integral \( \int \frac{\ln(10)}{u} du \) is a basic logarithmic integral. The result is \( \ln(10) \ln |u| + C \), where \( C \) is the constant of integration.
5Step 5: Back-Substitute for Original Variable
Replace \( u \) with \( \log_{10} x \) to return to the original variable: \[ \ln(10) \ln |\log_{10} x| + C. \] This is the solution to the original integral.
Key Concepts
Logarithmic SubstitutionBasic Logarithmic IntegralConstant of Integration
Logarithmic Substitution
Logarithmic substitution is a method used to simplify the integration process, especially when the integrand features both a variable and its logarithm.
In the exercise provided, the integral involves \( \log_{10} x\), to make this easier, we use the substitution \( u = \log_{10} x \).
This allows us to transform the integral into a much simpler form, making it easier to solve. Instead of dealing directly with the tricky combination of \( x \) and \( \log_{10} x \) in the denominator, we work with the substitution variable \( u \).
**Steps for Logarithmic Substitution**
* Identify that a logarithmic substitution is needed, often when you see a logarithmic function involved in a more complex integrand.
* Assign the logarithmic function to a new variable \( u\), which in our case is \( u = \log_{10} x \).
* Solve for \(dx\) in terms of \(du\) using the derivative of the logarithmic function.
* Substitute all parts of the integral with expressions involving \(u\), simplifying the integral.
This approach helps bypass algebraic difficulty, leading to a form that is readily integrable using basic techniques.
In the exercise provided, the integral involves \( \log_{10} x\), to make this easier, we use the substitution \( u = \log_{10} x \).
This allows us to transform the integral into a much simpler form, making it easier to solve. Instead of dealing directly with the tricky combination of \( x \) and \( \log_{10} x \) in the denominator, we work with the substitution variable \( u \).
**Steps for Logarithmic Substitution**
* Identify that a logarithmic substitution is needed, often when you see a logarithmic function involved in a more complex integrand.
* Assign the logarithmic function to a new variable \( u\), which in our case is \( u = \log_{10} x \).
* Solve for \(dx\) in terms of \(du\) using the derivative of the logarithmic function.
* Substitute all parts of the integral with expressions involving \(u\), simplifying the integral.
This approach helps bypass algebraic difficulty, leading to a form that is readily integrable using basic techniques.
Basic Logarithmic Integral
Once the substitution is made, the integral becomes a basic logarithmic integral. This means it has a straightforward solution where the integrand is of the form \( \frac{1}{u} \).
With the substitution \( u = \log_{10} x \), the integral simplifies to \( \int \frac{\ln(10)}{u} \, du \). Here, \( \ln(10) \) becomes a constant factor and can be taken out of the integral sign.
**Evaluating Basic Logarithmic Integral**
* The integral of \( \frac{1}{u} \) with respect to \( u \) is simply \( \ln |u| + C \), where \( C \) is an integration constant.
* In this case, multiplying outside the integral by \( \ln(10) \) gives the result \( \ln(10) \ln |u| + C \).
Basic logarithmic integrals are significant in calculus because they are among the fundamental and most straightforward antiderivatives, especially when derived through substitution.
With the substitution \( u = \log_{10} x \), the integral simplifies to \( \int \frac{\ln(10)}{u} \, du \). Here, \( \ln(10) \) becomes a constant factor and can be taken out of the integral sign.
**Evaluating Basic Logarithmic Integral**
* The integral of \( \frac{1}{u} \) with respect to \( u \) is simply \( \ln |u| + C \), where \( C \) is an integration constant.
* In this case, multiplying outside the integral by \( \ln(10) \) gives the result \( \ln(10) \ln |u| + C \).
Basic logarithmic integrals are significant in calculus because they are among the fundamental and most straightforward antiderivatives, especially when derived through substitution.
Constant of Integration
The constant of integration, denoted as \( C \), is essential in indefinite integrals. This concept reflects the general form of an antiderivative that includes not just one, but a family of functions.
When working with indefinite integrals, it's crucial to include \( C \), as it represents all possible vertical shifts of the antiderivative curve.
**Why We Include the Constant of Integration**
* Every indefinite integral represents a collection of functions differing only by a constant. Without \( C \), you'd miss versions of the function that could fit different initial conditions.
* Including \( C \) ensures that the solution is valid for the entire family of antiderivatives, accommodating any starting point value.
In our exercise, after integrating and back-substituting for \( u \), the final expression is \( \ln(10) \ln |\log_{10} x| + C \). This addition accounts for all potential functions that solve our integral, thus reinforcing its completeness.
When working with indefinite integrals, it's crucial to include \( C \), as it represents all possible vertical shifts of the antiderivative curve.
**Why We Include the Constant of Integration**
* Every indefinite integral represents a collection of functions differing only by a constant. Without \( C \), you'd miss versions of the function that could fit different initial conditions.
* Including \( C \) ensures that the solution is valid for the entire family of antiderivatives, accommodating any starting point value.
In our exercise, after integrating and back-substituting for \( u \), the final expression is \( \ln(10) \ln |\log_{10} x| + C \). This addition accounts for all potential functions that solve our integral, thus reinforcing its completeness.
Other exercises in this chapter
Problem 69
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