Textbook Question
76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x
44
views
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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.
Recommended video:
05:43
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.
Recommended video:
06:21Properties of Functions
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.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
85. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. More than one integration method can be used to evaluate ∫ (1 / (1 - x²)) dx.
46
views
Textbook Question
Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.
b. Use the substitution x = u² and the fact that ∫ from 0 to ∞ of e^(-u²) du = √(π/2) to show that Γ(1/2) = √π.
121
views
Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
a. Find a Midpoint Rule approximation to ∫[1 to 6] √(x³ + 1) dx using n = 50 subintervals.
38
views
Textbook Question
Area and volume Consider the function f(x) = (9 + x²)^(-1/2) and the region R on the interval [0, 4] (see figure).
b. Find the volume of the solid generated when R is revolved about the x-axis.
62
views
Textbook Question
68. Different methods
b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.
70
views