Explanation Given Information: The rate of sale of paint-by-number sets is s ( t ) = 900 e 0.25 t 3 + e 0.25 t sets per month. Formula used: Follow the steps as mentioned below to solve an integral ∫ f d x by using substitution rule. Step1: Assume a function u in terms of x from f . Step2: Take the derivative d u d x of the function obtained in step 1 and solve for d x in terms of d u . Step3: Substitute the value of u from step 1 and the value of d x from step 2 in the provided integral and then solve the integral. Calculation: Consider that the sale in the starting, so, S ( 0 ) = 0 . Also consider that rate of sale of paint-by-number sets is s ( t ) = 900 e 0.25 t 3 + e 0.25 t sets per month. Total sale of paint-by-number sets S ( t ) is given by, S ( t ) = ∫ s ( t ) d t Substitute s ( t ) = 900 e 0.25 t 3 + e 0.25 t , S ( t ) = ∫ 900 e 0.25 t 3 + e 0.25 t d t Now evaluate the integral by using substitution rule. Assume a function u in terms of x from 900 e 0.25 t 3 + e 0.25 t , u = 3 + e 0.25 t Take the derivative d u d t of the function u = 3 + e 0.25 t , d u d t = 0.25 e 0.25 t Solve for d t , d t = 1 0.25 e 0.25 t d u Now substitute u = 3 + e 0.25 t and d t = 1 0.25 e 0.25 t d u in the integral ∫ 900 e 0.25 t 3 + e 0.25 t d t , ∫ 900 e 0.25 t 3 + e 0.25 t d t = ∫ 900 e 0.25 t u ⋅ 1 0.25 e 0.25 t d u = ∫ 3600 e 0.25 t u ⋅ 1 e 0.25 t d u Simplify the integral by canceling e 0.25 t . ∫ 3600 e 0.25 t u ⋅ 1 e 0.25 t d u = ∫ 3600 u d u = ∫ 3600 u d u Rewrite the integral ∫ 3600 u d u in power form as, ∫ 3600 u d u = ∫ 3600 u − 1 d u Evaluate the integral ∫ 3600 u − 1 d u by using constant multiple rule ∫ k ⋅ f ( x ) d x = k ∫ f ( x ) d x where k is a constant and the special case of the power rule for indefinite integral ∫ x − 1 d x = ln | x | + C when n = − 1

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ISBN: 9781337291248

Author: Waner, Stefan.

Publisher: Cengage Learning,

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Chapter 6.2, Problem 93E

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To calculate: The total sale of paint-by-number sets S(t) in t months and also calculate first 12 month sale, if the rate of sale of paint-by-number sets is s(t)=900e0.25t3+e0.25t sets per month.

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Chapter 6.2, Problem 92E

Chapter 6.2, Problem 94E

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The total sale of paint-by-number sets S ( t ) in t months and also calculate first 12 month sale, if the rate of sale of paint-by-number sets is s ( t ) = 900 e 0.25 t 3 + e 0.25 t sets per month.

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To calculate: The total sale of paint-by-number sets S(t) in t months and also calculate first 12 month sale, if the rate of sale of paint-by-number sets is s(t)=900e0.25t3+e0.25t sets per month.

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