Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.3.103

Chapter 7, Problem 7.3.103

Evaluate the integrals in Exercises 97–110.
103. ∫₁⁴ (ln 2 · log₂x / x) dx

Verified step by step guidance

Recognize that the integral is \( \int_1^4 \frac{\ln 2 \cdot \log_2 x}{x} \, dx \). Notice that \( \ln 2 \) is a constant and \( \log_2 x \) is the logarithm base 2 of \( x \).

Recall the change of base formula for logarithms: \( \log_2 x = \frac{\ln x}{\ln 2} \). Substitute this into the integral to rewrite the integrand in terms of natural logarithms.

After substitution, the integrand becomes \( \frac{\ln 2 \cdot \frac{\ln x}{\ln 2}}{x} = \frac{\ln x}{x} \). This simplifies the integral to \( \int_1^4 \frac{\ln x}{x} \, dx \).

To solve \( \int \frac{\ln x}{x} \, dx \), use the substitution \( t = \ln x \), which implies \( dt = \frac{1}{x} dx \). This transforms the integral into \( \int t \, dt \).

Integrate \( \int t \, dt \) to get \( \frac{t^2}{2} + C \). Substitute back \( t = \ln x \) to express the antiderivative as \( \frac{(\ln x)^2}{2} + C \). Finally, evaluate this expression at the limits \( x=1 \) and \( x=4 \) to find the definite integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Change of Logarithm Base

The integral involves logarithms with different bases (natural log and base 2). Converting all logarithms to a common base, typically the natural logarithm, simplifies the expression and makes integration manageable. Use the formula log_a(x) = ln(x) / ln(a) to rewrite log₂(x) in terms of ln(x).

Properties of Logarithms

Understanding logarithm properties, such as the product, quotient, and power rules, helps simplify the integrand. In this problem, recognizing that ln(2)·log₂(x) can be expressed as ln(x) is key to reducing the integral to a simpler form that is easier to integrate.

Integration of Logarithmic Functions

Integrating functions involving logarithms often requires techniques like substitution or integration by parts. For example, integrating (ln x)/x dx can be done by recognizing it as the derivative of (ln x)^2 / 2. Familiarity with these methods is essential to evaluate the integral correctly.

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25. First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-gluconolactone into gluconic acid, for example,

dy/dt = -0.6y

when t is measured in hours. If there are 100 grams of δ-gluconolactone present when t=0, how many grams will be left after the first hour?

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