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Magazine Chapter 7: Problem 45
Evaluate the integrals. $$\int \frac{d x}{x \log _{10} x}$$
Short Answer
Expert verified
The integral evaluates to \( \ln(10) \ln|\log_{10} x| + C \).
Step by step solution
01
Identify the substitution
Observe that the derivative of \( \log_{10} x \) can simplify the integration process. To tackle this, let's use the substitution method. Set \( u = \log_{10} x \).
02
Differentiate and substitute
Differentiate \( u \) with respect to \( x \) to find \( du \). Recall that \( \frac{d}{dx} \log_{10} x = \frac{1}{x \ln(10)} \). Therefore, \( du = \frac{1}{x \ln(10)} dx \) and rearrange to get \( dx = x \ln(10) \, du \).
03
Simplify the integral
Substitute \( u = \log_{10} x \) and \( dx = x \ln(10) \, du \) into the integral. Thus, it becomes \( \int \frac{x \ln(10) \, du}{x u} \), which simplifies to \( \int \frac{\ln(10)}{u} du \).
04
Integrate
The simplified integral \( \int \frac{\ln(10)}{u} \, du \) is a basic logarithmic form. Integrate to get \( \ln(10) \ln|u| + C \).
05
Back-substitute
Substitute back for \( u \) to express the result in terms of \( x \). Since \( u = \log_{10} x \), the integral becomes \( \ln(10) \ln|\log_{10} x| + C \).
06
Final Expression
Thus, the integral \( \int \frac{d x}{x \log_{10} x} = \ln(10) \ln|\log_{10} x| + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
One popular technique for solving integrals is the substitution method. It transforms a complicated integral into an easier form by substituting part of the integral with a single variable.
To apply the substitution method effectively,
This choice works well because the derivative of \( u \), which is related to \( x \), appears in the integrand, allowing us to simplify the expression after substitution.
The method requires practice to identify the right substitution, but it is a fundamental skill in integration.
To apply the substitution method effectively,
- Choose a substitution that simplifies the integrand, often by setting a part of it equal to a new variable.
- Differentiate the chosen substitution to express the differential aspect of the integral in terms of the new variable.
- Replace all occurrences of the original variable in the integral with expressions involving the new variable.
This choice works well because the derivative of \( u \), which is related to \( x \), appears in the integrand, allowing us to simplify the expression after substitution.
The method requires practice to identify the right substitution, but it is a fundamental skill in integration.
Logarithmic Integration
Logarithmic integration deals with integrals of the form \( \int \frac{1}{u} \, du \). This form is special because it results in a logarithmic function.
It's one of the basic integral forms where:
This is easily solved using the logarithmic integration rule to obtain \( \ln(10) \ln|u| + C \).
Logarithmic integration is a valuable technique when functions within the integrand lead directly to a natural log or need to be converted into one with a substitution.
It's one of the basic integral forms where:
- The integral \( \int \frac{1}{u} \, du \) results in \( \ln |u| + C \), where \( C \) is the constant of integration.
This is easily solved using the logarithmic integration rule to obtain \( \ln(10) \ln|u| + C \).
Logarithmic integration is a valuable technique when functions within the integrand lead directly to a natural log or need to be converted into one with a substitution.
Differentiation
Differentiation plays a crucial role in integration, especially when verifying the solution or simplifying integrals. In essence, differentiation is used to break down complex expressions into simpler components.
Knowing that \( \frac{d}{dx} \log_{10} x = \frac{1}{x \ln(10)} \) was essential to find the appropriate equivalent expression for \( dx \) during substitution.
Thus, while integration focuses on finding the area under a curve, it heavily relies on differentiation principles to manipulate and simplify the integrand.
- It allows us to express one variable in terms of another, a key step in substitution.
- Understanding differentiation rules ensures that substitutions are correctly chosen and applied.
Knowing that \( \frac{d}{dx} \log_{10} x = \frac{1}{x \ln(10)} \) was essential to find the appropriate equivalent expression for \( dx \) during substitution.
Thus, while integration focuses on finding the area under a curve, it heavily relies on differentiation principles to manipulate and simplify the integrand.
