Section 8.2 Calculus BC AP/Dual, Revised ©2019 viet.dang@humbleisd.net Integration by Parts Section 8.2 Calculus BC AP/Dual, Revised ©2019 viet.dang@humbleisd.net 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Review Product Rule: Integration of Product Rule 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Integrate by Parts 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Used to simplify integrals of the form, 𝒇 𝒙 𝒈 𝒙 𝒅𝒙 These products include algebraic and transcendental functions, such as inverse, logarithmic, trigonometric, and exponential functions Used to assist in taking the integral of a complicated problem Equation: 𝒖𝒅𝒗 =𝒖𝒗− 𝒗𝒅𝒖 “ultra-violet voo-doo” 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Steps Determine the appropriate substitution for 𝒖 and 𝒅𝒗 where 𝒖 is the most simple and 𝒅𝒗 is the most complicated form To Establish 𝒖, “𝑳𝑰𝑨𝑻𝑬 for 𝑼” is BASED ON ORDER TO DETERMINE WHAT IS “𝑼” Logarithmic Inverse trigonometric Algebraic Trigonometric Exponential Integrate 𝒅𝒗 to solve for 𝒗, and differentiate 𝒖 to solve for 𝒅𝒖. Substitute the values for 𝒖, 𝒗, 𝒅𝒖, and 𝒅𝒗 into the formula and solve 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts LIATE Examples To Establish 𝒖, “𝑳𝑰𝑨𝑻𝑬 for 𝑼” is BASED ON ORDER TO DETERMINE WHAT IS “𝑼” Logarithmic 𝐥𝐧 𝒙 , 𝐥𝐨𝐠 𝟐 𝒙 ,𝒆𝒕𝒄… Inverse trigonometric 𝐭𝐚𝐧 −𝟏 𝒙 , 𝐚𝐫𝐜𝐬𝐢𝐧 𝒙 ,𝒆𝒕𝒄… Algebraic 𝒙, 𝒙 𝟐 ,𝒙+𝟏,𝒆𝒕𝒄… Trigonometric 𝐜𝐨𝐬 𝒙 ,𝐭𝐚𝐧(𝒙),𝒆𝒕𝒄… Exponential 𝒆 𝒙 , 𝟐 𝒙 ,𝒆𝒕𝒄… 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Example 1 Solve 𝒙 𝒆 𝟐𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Example 1 Solve 𝒙 𝒆 𝟐𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Example 2 Solve 𝒙 𝐥𝐧𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Example 2 Solve 𝒙 𝐥𝐧𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Your Turn Solve 𝒙 𝟒 𝐥𝐧𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Example 3 Solve 𝟒 𝐚𝐫𝐜𝐜𝐨𝐬𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Example 3 Solve 𝟒 𝐚𝐫𝐜𝐜𝐨𝐬𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Example 4 Solve 𝒆 𝒙 𝐜𝐨𝐬𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Your Turn Solve 𝒙 𝟐 𝐜𝐨𝐬𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Example 5 Solve 𝐥𝐧 𝒙 𝒅𝒙 2/28/2019 3:41 PM §8.2: Integration by Parts

AP Multiple Choice Practice Question 1 (non-calculator) Solve 𝒙 𝟐 𝐜𝐨𝐬 𝒙 𝒅𝒙=𝒇 𝒙 − 𝟐𝒙 𝐬𝐢𝐧 𝒙 𝒅𝒙, then 𝒇 𝒙 = (A) 𝟐 𝐬𝐢𝐧 𝒙 + 𝟐𝒙 𝐜𝐨𝐬𝒙+𝑪 (B) 𝒙 𝟐 𝐬𝐢𝐧 𝒙+𝑪 (C) 𝟐𝒙 𝐜𝐨𝐬 𝒙− 𝒙 𝟐 𝐬𝐢𝐧 𝒙+𝑪 (D) 𝟐− 𝒙 𝟐 𝐜𝐨𝐬 𝒙−𝟒 𝐬𝐢𝐧 𝒙+𝑪 2/28/2019 3:41 PM §8.2: Integration by Parts

AP Multiple Choice Practice Question 1 (non-calculator) Solve 𝒙 𝟐 𝐜𝐨𝐬 𝒙 𝒅𝒙=𝒇 𝒙 − 𝟐𝒙 𝐬𝐢𝐧 𝒙 𝒅𝒙, then 𝒇 𝒙 = Vocabulary Connections and Process Answer 2/28/2019 3:41 PM §8.2: Integration by Parts

§8.2: Integration by Parts Assignment Page 521 11-19 odd, 23-27 odd, 31, 32 2/28/2019 3:41 PM §8.2: Integration by Parts