Domanda del libro di testo
4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.
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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 8.8.81a
81. Find the values of p for which each integral converges.
a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]
Guida verificata passo dopo passo1
Identify the integral and the parameter involved: we have the integral \( \int_1^2 \frac{dx}{x (\ln x)^p} \), and we want to find for which values of \( p \) this integral converges.
Check the behavior of the integrand near the limits of integration. Since the interval is from 1 to 2, and \( \ln 1 = 0 \), the potential issue is at the lower limit \( x = 1 \) where the denominator involves \( (\ln x)^p \) which tends to zero, possibly causing a singularity.
Make a substitution to analyze the behavior near \( x = 1 \): let \( t = \ln x \). Then \( dt = \frac{1}{x} dx \), so \( dx = x dt = e^t dt \). The integral limits change from \( x=1 \) to \( t=0 \), and from \( x=2 \) to \( t=\ln 2 \).
Rewrite the integral in terms of \( t \): \( \int_0^{\ln 2} \frac{e^t dt}{e^t t^p} = \int_0^{\ln 2} \frac{dt}{t^p} \). Now the integral simplifies to \( \int_0^{\ln 2} t^{-p} dt \).
Determine convergence of \( \int_0^{a} t^{-p} dt \) for some positive \( a \). This integral converges if and only if \( -p > -1 \), or equivalently \( p < 1 \). So the original integral converges for \( p < 1 \).
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Improper Integrals and Convergence
An improper integral involves limits where the integrand becomes unbounded or the interval is infinite. To determine convergence, we analyze the behavior of the integrand near points causing potential issues, such as where the denominator approaches zero or infinity.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Behavior of the Natural Logarithm Function near 1
The natural logarithm ln(x) approaches 0 as x approaches 1 from the right. Since the integral's denominator includes (ln x)^p, understanding how ln(x) behaves near 1 is crucial to assess whether the integral converges or diverges.
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05:18Derivative of the Natural Logarithmic Function
p-Test for Convergence of Integrals
The p-test helps determine convergence of integrals of the form ∫ (1 / t^p) dt near zero or infinity. By substituting t = ln(x), the integral can be analyzed using the p-test to find values of p for which the integral converges.
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Percorso guidato
07:51Choosing a Convergence Test
Pratica correlata
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Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.
a. About the y-axis.
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Domanda del libro di testo
Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
a. u = 1/(x + 1)
What is the value of the integral?
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Domanda del libro di testo
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -2 to 0 of (x² - 1) dx
30
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Domanda del libro di testo
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 1 to 2 of 1/s² ds
30
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Domanda del libro di testo
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx
30
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