Domanda del libro di testo
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.
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.1.76b
76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x
Guida verificata passo dopo passo1
Step 1: Begin by substituting u = √x. This implies that x = u² and dx = 2u du. Rewrite the integral in terms of u.
Step 2: Substitute x = u² into the expression under the square root: √(x - x²) becomes √(u² - u⁴).
Step 3: Factorize the expression under the square root: u² - u⁴ = u²(1 - u²). Thus, √(u² - u⁴) = u√(1 - u²).
Step 4: Replace dx with 2u du and simplify the integral: ∫(1/√(x - x²)) dx becomes ∫(1/(u√(1 - u²)) * 2u du). Cancel out u in the numerator and denominator.
Step 5: The integral simplifies to ∫(2/√(1 - u²)) du. Recognize this as the standard integral for the arcsine function, ∫(1/√(1 - u²)) du = sin⁻¹(u) + C. Substitute back u = √x to get the final result: 2 sin⁻¹√x + C.
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Substitution in Integration
Substitution is a technique used in integration to simplify the integrand. By replacing a variable with another expression, the integral can often be transformed into a more manageable form. In this case, substituting u = √x allows us to express the integral in terms of u, making it easier to evaluate.
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Substitution With an Extra Variable
Inverse Trigonometric Functions
Inverse trigonometric functions, such as sin⁻¹(x), are used to find angles when given the value of a trigonometric function. In this problem, the result involves sin⁻¹(√x), which indicates that the integral's solution relates to the angle whose sine is √x. Understanding these functions is crucial for interpreting the final result of the integral.
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06:35Derivatives of Other Inverse Trigonometric Functions
Definite vs. Indefinite Integrals
Indefinite integrals represent a family of functions and include a constant of integration (C) in their results. The integral in this question is indefinite, meaning it does not have specified limits of integration. Recognizing the difference between definite and indefinite integrals is essential for correctly interpreting the outcome of integration problems.
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Percorso guidato
05:43Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo
93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:
vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).
b. Which car travels farthest on the interval 0 ≤ t ≤ 5?
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Domanda del libro di testo
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.
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Domanda del libro di testo
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.
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Domanda del libro di testo
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.
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Domanda del libro di testo
68. Different methods
b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.
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