Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (x² - 1)^(5/2), where x > 1
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Ch. 8 - Techniques of Integration
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 8, Problema 8.2.36
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (ln x)³/x dx
Guida verificata passo dopo passo1
Recognize that the integral is \( \int \frac{(\ln x)^3}{x} \, dx \). Notice that the integrand involves \( (\ln x)^3 \) divided by \( x \), which suggests a substitution related to \( \ln x \).
Use the substitution \( u = \ln x \). Then, the differential \( du = \frac{1}{x} dx \), which means \( dx = x \, du \). Since \( \frac{1}{x} dx = du \), the integral can be rewritten in terms of \( u \).
Rewrite the integral using the substitution: \( \int (\ln x)^3 \cdot \frac{1}{x} dx = \int u^3 \, du \). This simplifies the integral to a basic power function of \( u \).
Integrate \( u^3 \) with respect to \( u \) using the power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where here \( n = 3 \).
After integrating, substitute back \( u = \ln x \) to express the answer in terms of \( x \). This completes the evaluation of the integral.
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Integration by Substitution
Integration by substitution simplifies integrals by changing variables to make the integral easier to solve. For example, setting u = ln(x) transforms the integral into a polynomial form in terms of u, which is easier to integrate.
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04:27
Substitution With an Extra Variable
Properties of Logarithmic Functions
Understanding the natural logarithm function ln(x) and its derivative is crucial. Since d/dx [ln(x)] = 1/x, this relationship helps identify substitution candidates and simplifies integrals involving ln(x).
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Percorso guidato
06:21Properties of Functions
Integration of Polynomial Functions
Once substitution reduces the integral to a polynomial form, integrating powers of a variable (like u³) involves applying the power rule for integration: ∫u^n du = u^(n+1)/(n+1) + C, which is straightforward and essential for solving the problem.
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07:00Taylor Polynomials
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