Section 4.5A Calculus AP/Dual, Revised ©2019 viet.dang@humbleisd.net Change of Variables Section 4.5A Calculus AP/Dual, Revised ©2019 viet.dang@humbleisd.net 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Use a Change of Variables to evaluate a definite integral When applying 𝒖-substitution to evaluate a definite integral, make sure to change the Limits of Integration before evaluating the integral If the function 𝒖=𝒈(𝒙) has a continuous derivative on the closed interval 𝒂, 𝒃 and f is continuous on the range of g, then: 𝒂 𝒃 𝒇(𝒈(𝒙))𝒈′(𝒙)𝒅𝒙= 𝒈 𝒂 𝒈 𝒃 𝑭(𝒈(𝒙))+𝑪 𝒂 = Lower Bound, 𝒃 = Upper Bound 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Steps Simplify/Rewrite the problem Establish the 𝒖 as the base, 𝒅𝒖 as the derivative, and 𝒅𝒙 as the constant outside Integrate: “Add and Divide” Simplify Apply 𝒃 and 𝒂 into the 𝒖-substitution equation because of the new limits and subtract 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 1 Solve 𝟎 𝟏 𝒙 𝒙 𝟐 +𝟏 𝟑 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 1 Solve 𝟎 𝟏 𝒙 𝒙 𝟐 +𝟏 𝟑 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 2 Solve 𝟎 𝟏 𝒙 𝟐 𝒙 𝟑 +𝟖 𝟐 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Your Turn Solve 𝟏 𝟓 𝟐𝒙−𝟏 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 3 Solve 𝟏 𝟓 𝒙 𝟐𝒙−𝟏 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 3 Solve 𝟏 𝟓 𝒙 𝟐𝒙−𝟏 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 3 Solve 𝟏 𝟓 𝒙 𝟐𝒙−𝟏 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 4 Solve 𝟎 𝟒 𝟏 𝟐𝒙+𝟏 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Your Turn Solve 𝟎 𝟐 𝒙 𝟏+𝟐 𝒙 𝟐 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 5 Solve −𝝅 𝝅 𝟐 𝐜𝐨𝐬 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 6 Solve 𝟎 𝝅/𝟐 𝟒 𝐬𝐢𝐧 𝒙 𝟐 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Your Turn Solve 𝟎 𝝅/𝟒 𝟒 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 7 Solve 𝟎 𝟖 𝒆 −𝟒𝒙 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Example 8 Solve 𝟎 𝟏 𝟐 𝟏+ 𝒙 𝟐 𝒅𝒙 using change of variables 2/18/2019 6:04 PM §4.5A: Change of Variables

AP Multiple Choice Question 1 (non-calculator) Using the substitution, 𝒖= 𝒙 , 𝟏 𝟒 𝒆 𝒙 𝒙 𝒅𝒙 is equal to the following? 𝟐 𝟏 𝟒 𝒆 𝒖 𝒅𝒖 𝟐 𝟏 𝟐 𝒆 𝒖 𝒅𝒖 𝟏 𝟐 𝟏 𝟐 𝒆 𝒖 𝒅𝒖 𝟏 𝟒 𝒆 𝒖 𝒅𝒖 2/18/2019 6:04 PM §4.5A: Change of Variables

AP Multiple Choice Question 1 (non-calculator) Using the substitution, 𝒖= 𝒙 , 𝟏 𝟒 𝒆 𝒙 𝒙 𝒅𝒙 is equal to the following? Vocabulary Connections and Process Answer and Justifications 2/18/2019 6:04 PM §4.5A: Change of Variables

§4.5A: Change of Variables Assignment Worksheet 2/18/2019 6:04 PM §4.5A: Change of Variables