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Evaluate the definite integral. ra2/9 x° + a² dx (a > 0) 3 20 27

Evaluate the definite integral. ra2/9 x° + a² dx (a > 0) 3 20 27

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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### Integral Evaluation Challenge Evaluate the definite integral: \[ \int_{0}^{a^{2/9}} x^8 \sqrt[9]{x^9 + a^2} \, dx \quad (a > 0) \] ### Solution Attempt The solution provided in the image is \[ \frac{20^3}{27} \left( 2 \left( \frac{3}{2} \right) - 1 \right) \] The red cross indicates that this solution is incorrect. ### Explanation - The problem involves finding the definite integral of the function \( x^8 \sqrt[9]{x^9 + a^2} \) from 0 to \( a^{2/9} \). - The variable \( a \) is a positive real number, as indicated by \( a > 0 \). ### Note This integral requires careful integration techniques, possibly involving substitutions or transformations, to be correctly evaluated. The incorrect solution suggests a mathematical mistake was made in the integration process.
Transcribed Image Text:### Integral Evaluation Challenge Evaluate the definite integral: \[ \int_{0}^{a^{2/9}} x^8 \sqrt[9]{x^9 + a^2} \, dx \quad (a > 0) \] ### Solution Attempt The solution provided in the image is \[ \frac{20^3}{27} \left( 2 \left( \frac{3}{2} \right) - 1 \right) \] The red cross indicates that this solution is incorrect. ### Explanation - The problem involves finding the definite integral of the function \( x^8 \sqrt[9]{x^9 + a^2} \) from 0 to \( a^{2/9} \). - The variable \( a \) is a positive real number, as indicated by \( a > 0 \). ### Note This integral requires careful integration techniques, possibly involving substitutions or transformations, to be correctly evaluated. The incorrect solution suggests a mathematical mistake was made in the integration process.
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