(c) Observe that the examples in (b) work nicely because of the derivatives you were asked to calculate in (a). Each integrand in (b) is precisely the result of differentiating one of the products of basic functions found in (a). To see what happens when an integrand is still a product but not necessarily the result of differentiating an elementary product, we consider how to evaluate x cos(x) dx. (i) First, observe that d -[x sin(x)] = x cos(x) + sin(x). dx Integrating both sides indefinitely and using the fact that the integral of a sum is the sum of the integrals, we find that d [x sin(x)] ) dx x cos(x) dx + sin(x) dx. dx In this last equation, evaluate the indefinite integral on the left side: S( sin(x)]) dæ = -XCOSX + Sinx + C Now evaluate the indefinite integral on the right side:
(c) Observe that the examples in (b) work nicely because of the derivatives you were asked to calculate in (a). Each integrand in (b) is precisely the result of differentiating one of the products of basic functions found in (a). To see what happens when an integrand is still a product but not necessarily the result of differentiating an elementary product, we consider how to evaluate x cos(x) dx. (i) First, observe that d -[x sin(x)] = x cos(x) + sin(x). dx Integrating both sides indefinitely and using the fact that the integral of a sum is the sum of the integrals, we find that d [x sin(x)] ) dx x cos(x) dx + sin(x) dx. dx In this last equation, evaluate the indefinite integral on the left side: S( sin(x)]) dæ = -XCOSX + Sinx + C Now evaluate the indefinite integral on the right side:
Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015
1st Edition
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:HOUGHTON MIFFLIN HARCOURT
Chapter8: Graphing Quadratic Functions
Section: Chapter Questions
Problem 17CT
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part C
(i) and (iii)
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