Textbook Question
Evaluating integrals Evaluate the following integrals.
∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍
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8. Definite Integrals
Introduction to Definite Integrals
2:29 minutesProblem 5.R.71
Textbook Question
Evaluating integrals Evaluate the following integrals.
∫₋₅⁵ ω³ /√(ω⁵⁰ + ω²⁰ + 1) dω (Hint: Use symmetry . )
Verified step by step guidance1
Recognize that the integral has symmetric limits of integration (-5 to 5) and analyze the integrand for symmetry. Specifically, check if the function is odd or even. A function f(ω) is odd if f(-ω) = -f(ω), and even if f(-ω) = f(ω).
Substitute -ω into the integrand to test for symmetry. The numerator ω³ changes sign to (-ω)³ = -ω³, while the denominator √(ω⁵⁰ + ω²⁰ + 1) remains unchanged because all powers of ω in the denominator are even.
Conclude that the integrand is an odd function because the numerator changes sign while the denominator does not. For an odd function integrated over symmetric limits (-a to a), the integral evaluates to 0.
Use the property of definite integrals for odd functions over symmetric intervals: ∫₋ₐᵃ f(ω) dω = 0. Apply this property to the given integral.
State that the integral evaluates to 0 due to the symmetry of the integrand and the odd function property over symmetric limits.
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