Domanda del libro di testo
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
35. y = ln((x²+1)^5/√(1-x))
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.3.103
Evaluate the integrals in Exercises 97–110.
103. ∫₁⁴ (ln 2 · log₂x / x) dx
Guida verificata passo dopo passo1
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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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).
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05:36
Change of Base Property
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.
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05:36Change of Base Property
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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5:26Graphs of Logarithmic Functions
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