Textbook Question
63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. If m is a positive integer, then ∫[0 to π] sin^m(x) dx = 0.
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.2.58b
58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:
b. Use substitution.
Verified step by step guidance1
Start with the integral \( \int_0^{\frac{\pi}{3}} \sin(x) \cdot \ln(\cos(x)) \, dx \). The goal is to use substitution to simplify this integral.
Choose the substitution \( u = \cos(x) \). Then, compute the differential: \( du = -\sin(x) \, dx \), which implies \( \sin(x) \, dx = -du \).
Rewrite the integral in terms of \( u \). When \( x = 0 \), \( u = \cos(0) = 1 \). When \( x = \frac{\pi}{3} \), \( u = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \). Substitute these limits and the expression for \( \sin(x) \, dx \) into the integral:
\[ \int_0^{\frac{\pi}{3}} \sin(x) \ln(\cos(x)) \, dx = \int_1^{\frac{1}{2}} \ln(u) (-du) = \int_{\frac{1}{2}}^{1} \ln(u) \, du \]
Now, the integral is \( \int_{\frac{1}{2}}^{1} \ln(u) \, du \), which can be evaluated using integration by parts or known formulas for \( \int \ln(u) \, du \).
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8mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution is a technique used to simplify integrals by changing variables. It involves choosing a substitution that transforms the integral into a more manageable form, often by letting a part of the integrand equal a new variable. This method is especially useful when the integral contains a composite function.
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Substitution With an Extra Variable
Properties of Logarithmic and Trigonometric Functions
Understanding the behavior and derivatives of logarithmic and trigonometric functions is essential. For example, knowing that the derivative of ln(cos(x)) involves -tan(x) helps in choosing an effective substitution. Familiarity with these functions aids in manipulating the integral and recognizing suitable substitutions.
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Definite Integrals and Limits of Integration
When performing substitution in definite integrals, it is important to adjust the limits of integration according to the new variable. This avoids the need to revert to the original variable after integration and ensures the integral is evaluated correctly over the specified interval.
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Related Practice
Textbook Question
The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.
The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]
(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project ""The exponential Eiffel Tower"")
b. With a = 0 and c = 2, find the equations of the lines tangent to both curves at x = 0
Textbook Question
Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.
b. Find the area of the region bounded by the graph of g and the x-axis on the interval [0,2].
Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
b. Calculate f''(x).
Textbook Question
82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.
b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.
Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
71. Let f(x) = √(sin x).
b. Find an upper bound on the absolute error in the estimate from part (a) using Theorem 8.1. (Hint: |f''''(x)| ≤ 1 on [1,2].)