Problem 66 - Thomas Calculus

Question

Evaluate the integrals. \(\int_{1 / 10}^{10} \frac{\log _{10}(10 x)}{x} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \( 2\ln(10) \).

Step 1: Understand the Integral

We need to evaluate the definite integral \( \int_{1/10}^{10} \frac{\log_{10}(10x)}{x} \, dx \). The bounds of integration are from \( 1/10 \) to \( 10 \). The integrand is a logarithmic function divided by \( x \).

Step 2: Change of Base for the Logarithm

Recall the logarithm change of base formula: \( \log_{10}(a) = \frac{\ln(a)}{\ln(10)} \). Therefore, \( \log_{10}(10x) = \frac{\ln(10x)}{\ln(10)} \). Substitute this into the integrand: \( \int_{1/10}^{10} \frac{\ln(10x)}{x \ln(10)} \, dx \).

Step 3: Simplify the Integral

Factor out the constant \( \frac{1}{\ln(10)} \) from the integral: \( \frac{1}{\ln(10)} \int_{1/10}^{10} \frac{\ln(10x)}{x} \, dx \).

Step 4: Split the Logarithm

Use the logarithmic identity \( \ln(10x) = \ln(10) + \ln(x) \). Substitute this back into the integral: \[ \frac{1}{\ln(10)} \int_{1/10}^{10} \frac{\ln(10) + \ln(x)}{x} \, dx \] It becomes: \[ \frac{1}{\ln(10)} \left( \int_{1/10}^{10} \frac{\ln(10)}{x} \, dx + \int_{1/10}^{10} \frac{\ln(x)}{x} \, dx \right) \]

Step 5: Evaluate the First Integral

The first integral is \( \int_{1/10}^{10} \frac{\ln(10)}{x} \, dx \). The constant \( \ln(10) \) can be factored out: \( \ln(10) \int_{1/10}^{10} \frac{1}{x} \, dx \). The integral of \( \frac{1}{x} \) is \( \ln(x) \). Evaluate: \[ \ln(10) \left[ \ln(x) \right]_{1/10}^{10} = \ln(10) \left( \ln(10) - \ln(1/10) \right) = \ln(10) \cdot 2 \ln(10) \]

Step 6: Evaluate the Second Integral

The second integral is \( \int_{1/10}^{10} \frac{\ln(x)}{x} \, dx \). Notice this is a definite integral of the form \( \int f(x) f'(x) \, dx \), but direct integration using integration by parts is more straightforward. Let \( u = \ln(x) \) and \( dv = \frac{1}{x} \, dx \), then \( du = \frac{1}{x} \, dx \) and \( v = \ln(x) \). Using integration by parts \( \int u \, dv = uv - \int v \, du \): \[ \left. \frac{1}{2}(\ln(x))^2 \right|_{1/10}^{10} = \frac{1}{2} \left( (\ln(10))^2 - (\ln(1/10))^2 \right) = \frac{1}{2}((\ln(10))^2 - (\ln(10))^2) = 0 \]

Step 7: Combine Results

Combine the results from the two integrals obtained in Steps 5 and 6: \[ \frac{1}{\ln(10)} (2 (\ln(10))^2 + 0) = 2 \ln(10) \] Therefore, the final evaluated integral is \( 2 \ln(10) \).

Step 8: Conclusion: Evaluate the Integral

Thus, the integral \( \int_{1/10}^{10} \frac{\log_{10}(10x)}{x} \, dx \) evaluates to \( 2 \ln(10) \).

Key Concepts

Logarithmic Functions

Logarithmic functions involve an equation or expression of the form \( \log_b(a) \), which represents the exponent or power that the base \( b \) must be raised to produce the number \( a \). Logarithms are the inverse operations of exponentiation, and they are important for simplifying multiplication and division in equations and for solving problems involving exponential growth or decay.

In this exercise, the logarithm \( \log_{10}(10x) \) is used. Here, base 10 is applied, which is commonly known as the common logarithm, and it's often used in scientific contexts.

Understanding the properties of logarithms is fundamental for manipulation and simplification. For example, the property \( \ln(ab) = \ln(a) + \ln(b) \) is crucial for splitting log terms to facilitate integration. This property was applied in splitting \( \ln(10x) \) into \( \ln(10) + \ln(x) \). This type of manipulation benefits the process of integration by breaking down complex expressions into simpler components.

Integration by Parts

Integration by parts is a powerful technique used in calculus to solve integrals of products of functions. It is derived from the product rule for differentiation and provides a method to integrate by transforming the original integral into another potentially simpler integral to solve.

The formula for integration by parts is given by:

• \( \int u \, dv = uv - \int v \, du \)

To effectively use this formula, we choose functions \( u \) and \( dv \) from the integrand, replacing them such that the resulting integral \( \int v \, du \) becomes simpler.

For the integral \( \int \frac{\ln(x)}{x} \, dx \), by setting \( u = \ln(x) \) and \( dv = \frac{1}{x} \, dx \), we follow the integration by parts method:

• \( du = \frac{1}{x} \, dx \)
• \( v = \ln(x) \)

These choices simplify the integral and allow us to find an expression for the integral that is easier to evaluate directly. This technique is useful for handling integrals where a straightforward anti-derivative is not apparent.

Change of Base Formula

The change of base formula for logarithms allows us to convert logarithmic expressions from one base to another. This conversion is particularly useful for simplifying expressions or when a specific base is more convenient for calculation.

The change of base formula is:

• \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \)

This property was employed in the given solution to transform \( \log_{10}(10x) \) into natural logarithms, as integration with the natural logarithm \( \ln \) is often more straightforward than with logarithms to other bases. The change to natural logarithms leverages the property \( \ln(ab) = \ln(a) + \ln(b) \), aiding in the simplification of the integrand.

Having this formula at hand is essential when working with different bases and needing to integrate or differentiate logarithmic terms. It helps align the problem into a more manageable form, reducing complexity and fostering a better understanding of how the expression behaves.