Explanation Formula used: For a constant C , ∫ x n d x = x n + 1 n + 1 + C , n ≠ − 1 (1) Calculation: Given integral is ∫ ( 1 − t 2 ) 3 t d t . Let x = ( 1 − t 2 ) . Differentiate on both sides, d x = − 2 t d t t d t = − 1 2 d x Substitute these values in the given integral and obtain as, ∫ ( 1 − t 2 ) 3 t d t = ∫ ( x ) 3 ( − 1 2 d x ) = − 1 2 ∫ ( x ) 3 d x Use formula (1) and obtain the integral of f ( x ) = x 3 as follows

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 5, Problem 4RE

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Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

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The integral value for ∫ ( 1 − t 2 ) 3 t d t .

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