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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.2.58b
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 guidance
1
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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Key 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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Properties of Functions

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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Definition of the Definite Integral
Related Practice
Textbook Question

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

46. ∫(0 to 2) x⁴ dx; n = 30

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

Textbook Question

2. Give an example of each of the following.

b. A repeated linear factor

Textbook Question

59. Two Methods

b. Evaluate ∫(x / √(x + 1)) dx using substitution.

1
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Textbook Question

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:

s(t) = e⁻ᵗ sin t

c. Generalize part (b) and find the average value of the position on the interval [nπ, (n+1)π], for n = 0, 1, 2, ...

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].)

Textbook Question

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.