Textbook Question
76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x
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 8.3.63b
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.
Verified step by step guidance1
Step 1: Understand the problem. The statement claims that the integral of sin^m(x) from 0 to π equals 0 for any positive integer m. We need to determine if this is true or false and provide an explanation or counterexample.
Step 2: Recall the properties of the sine function. The sine function, sin(x), is symmetric about π/2 within the interval [0, π]. This symmetry affects the behavior of sin^m(x) depending on whether m is odd or even.
Step 3: Analyze the case when m is odd. For odd values of m, sin^m(x) is an odd function about π/2. This means the integral from 0 to π will not necessarily be zero because the positive and negative contributions do not cancel out.
Step 4: Analyze the case when m is even. For even values of m, sin^m(x) is an even function about π/2. This symmetry ensures that the integral from 0 to π will be positive, as the function remains non-negative throughout the interval.
Step 5: Conclude that the statement is false. The integral ∫[0 to π] sin^m(x) dx is not always zero for positive integer values of m. Provide a counterexample, such as m = 2, where the integral evaluates to a positive value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral represents the signed area under a curve between two specified limits. In this case, the integral of sin^m(x) from 0 to π calculates the total area between the curve of sin^m(x) and the x-axis over that interval. Understanding how definite integrals work is crucial for evaluating the truth of the statement.
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Definition of the Definite Integral
Properties of the Sine Function
The sine function oscillates between -1 and 1, and its behavior over the interval [0, π] is particularly important. Specifically, sin(x) is non-negative in this interval, meaning that sin^m(x) will also be non-negative for any positive integer m. This property is essential for determining whether the integral can equal zero.
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Even and Odd Functions
An even function is symmetric about the y-axis, while an odd function is symmetric about the origin. The function sin^m(x) is even when m is even, and odd when m is odd. This distinction affects the evaluation of the integral, as the integral of an odd function over a symmetric interval around zero is zero, while the integral of an even function is positive, reinforcing the need to analyze the parity of m.
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Related Practice
Textbook Question
88. Incorrect Calculation
b. Evaluate ∫(from -1 to 1) dx/x or show that the integral does not exist.
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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.