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(e) [e*, 1 eartb dx Leartb + C a

(e) [e*, 1 eartb dx Leartb + C a

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 77E
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solve c d e

Question 4 Verify the following antidifferentiation formulas. (a) / (b) / In x dx = x ln x – x + C tan x dx = - In | cos a| + C COS 1 ar+b dx = ear+b + C a (d) / sin(ax + b) dæ 1 cos(ax + b) + C a (e) / 1 (a.x + b)"+1 + C (ax + b)" d.x = а(n + 1) The last three are cases of “baby substitution" the chain rule with a linear inside inside function ax + b establishes 1 | f'(a.x + b) da = =f(a.x + b) + C. d (ax+b) dx do not do this with more complicated This idea works because = a is a constant function inside functions! er? is not an antiderivative of e*- 2x For example, the quotient rule says that 2r (e*2x) d d.x 2.2 t ez? 2x 4x2 In fact, it is known that we will never obtain a “nice formula" for dx in the same way we can for 1 e2" + C. e2x d.x = 2
Transcribed Image Text:Question 4 Verify the following antidifferentiation formulas. (a) / (b) / In x dx = x ln x – x + C tan x dx = - In | cos a| + C COS 1 ar+b dx = ear+b + C a (d) / sin(ax + b) dæ 1 cos(ax + b) + C a (e) / 1 (a.x + b)"+1 + C (ax + b)" d.x = а(n + 1) The last three are cases of “baby substitution" the chain rule with a linear inside inside function ax + b establishes 1 | f'(a.x + b) da = =f(a.x + b) + C. d (ax+b) dx do not do this with more complicated This idea works because = a is a constant function inside functions! er? is not an antiderivative of e*- 2x For example, the quotient rule says that 2r (e*2x) d d.x 2.2 t ez? 2x 4x2 In fact, it is known that we will never obtain a “nice formula" for dx in the same way we can for 1 e2" + C. e2x d.x = 2
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