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(e) Find all values of a Є R such that the integral is convergent. L₁ f(x)(1 − x²)ª dr ⇒ f(x) = We observe that f is continuous and odd in (-1, 1). It follows that it suffices to discuss the behavior of the integral in the interval [0, 1]. For x 1 we have x3 x3 f(x) = 2.3 x3 √√√x4 – 1| - x4 √(1 − x)(1 + x)(1 + x²) 1 √√1 − x√√(1 + x)(1 + x²) 2(1-x)1/2 Hence 1 f(x)(1 − x²), - 2(1−z)1/2(1−z)^2@ and the integral converges if 1½-½ - α < 1, that is a > −1, and it diverges if 29-1 (1 − x) ź—a - a 1 that is a ≤ -12.

(e) Find all values of a Є R such that the integral is convergent. L₁ f(x)(1 − x²)ª dr ⇒ f(x) = We observe that f is continuous and odd in (-1, 1). It follows that it suffices to discuss the behavior of the integral in the interval [0, 1]. For x 1 we have x3 x3 f(x) = 2.3 x3 √√√x4 – 1| - x4 √(1 − x)(1 + x)(1 + x²) 1 √√1 − x√√(1 + x)(1 + x²) 2(1-x)1/2 Hence 1 f(x)(1 − x²), - 2(1−z)1/2(1−z)^2@ and the integral converges if 1½-½ - α < 1, that is a > −1, and it diverges if 29-1 (1 − x) ź—a - a 1 that is a ≤ -12.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(e) Find all values of a Є R such that the integral is convergent. L₁ f(x)(1 − x²)ª dr ⇒ f(x) = We observe that f is continuous and odd in (-1, 1). It follows that it suffices to discuss the behavior of the integral in the interval [0, 1]. For x 1 we have x3 x3 f(x) = 2.3 x3 √√√x4 – 1| - x4 √(1 − x)(1 + x)(1 + x²) 1 √√1 − x√√(1 + x)(1 + x²) 2(1-x)1/2 Hence 1 f(x)(1 − x²), - 2(1−z)1/2(1−z)^2@ and the integral converges if 1½-½ - α < 1, that is a > −1, and it diverges if 29-1 (1 − x) ź—a - a 1 that is a ≤ -12.
Transcribed Image Text:(e) Find all values of a Є R such that the integral is convergent. L₁ f(x)(1 − x²)ª dr ⇒ f(x) = We observe that f is continuous and odd in (-1, 1). It follows that it suffices to discuss the behavior of the integral in the interval [0, 1]. For x 1 we have x3 x3 f(x) = 2.3 x3 √√√x4 – 1| - x4 √(1 − x)(1 + x)(1 + x²) 1 √√1 − x√√(1 + x)(1 + x²) 2(1-x)1/2 Hence 1 f(x)(1 − x²), - 2(1−z)1/2(1−z)^2@ and the integral converges if 1½-½ - α < 1, that is a > −1, and it diverges if 29-1 (1 − x) ź—a - a 1 that is a ≤ -12.
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