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Ex 7.5, 18 - Integrate (x2 + 1) (x2 + 2) / (x3+3) (x2+4)

Ex 7.5, 18 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.5, 18 - Chapter 7 Class 12 Integrals - Part 3 Ex 7.5, 18 - Chapter 7 Class 12 Integrals - Part 4


Transcript

Ex 7.5, 18 Integrate the function (๐‘ฅ2+ 1)(๐‘ฅ2+ 2)/(๐‘ฅ2+ 3)(๐‘ฅ2+ 4) (๐‘ฅ^2 + 1)(๐‘ฅ^2 + 2)/(๐‘ฅ^2 + 3)(๐‘ฅ^2 + 4) " " Let t = ๐‘ฅ^2 = (๐‘ก + 1)(๐‘ก + 2)/(๐‘ก + 3)(๐‘ก + 4) = (๐‘ก^2 + 3๐‘ก + 2)/(๐‘ก^2 + 7๐‘ก + 12) = 1 + (โˆ’4๐‘ก โˆ’10)/(๐‘ก^2 + 7๐‘ก + 12) = 1 + (โˆ’(4๐‘ก + 10))/(๐‘ก + 3)(๐‘ก + 4) Rough = 1 โˆ’ ( (4๐‘ก + 10))/(๐‘ก + 3)(๐‘ก + 4) We can write (4๐‘ก + 10)/((๐‘ก + 3) (๐‘ก + 4) ) = ๐ด/((๐‘ก + 3) ) + ๐ต/((๐‘ก + 4) ) (4๐‘ก + 10)/((๐‘ก + 3) (๐‘ก + 4) ) = (๐ด(๐‘ก + 4) + ๐ต(๐‘ก + 3))/((๐‘ก + 3) (๐‘ก + 4) ) Cancelling denominator 4๐‘กโˆ’10 = ๐ด(๐‘ก+4)+๐ต(๐‘ก+3) Putting t = โˆ’ 4 in (1) 4(โˆ’4)+10 = ๐ด(โˆ’4+4)+๐ต(โˆ’4+3) โˆ’16+10 = ๐ดร—0+๐ต(โˆ’1) โ€ฆ(1) โˆ’6 = ๐ดร—0+๐ต(โˆ’1) โˆ’6 = โˆ’๐ต ๐ต = 6 Putting t = โˆ’3 in (1) 4๐‘กโˆ’10 = ๐ด(๐‘ก+4)+๐ต(๐‘ก+3) 4(โˆ’3)+10 = ๐ด(โˆ’3+4)+๐ต(โˆ’3+3) โˆ’12+10 = ๐ดร—1+๐ตร—0 โˆ’2 = ๐ด ๐ด = โˆ’2 Hence we can write (4๐‘ก + 10)/((๐‘ก + 3) (๐‘ก + 4) ) = (โˆ’2)/((๐‘ก + 3) ) + 6/((๐‘ก + 4) ) Putting back t = ๐‘ฅ^2 (4๐‘ฅ^2 โˆ’ 10)/((๐‘ฅ^2 + 3) (๐‘ฅ^2 + 4) ) = (โˆ’2)/((๐‘ฅ^2 + 3) ) + 6/((๐‘ฅ^2 + 4) ) Therefore โˆซ1โ–’(๐‘ฅ2+ 1)(๐‘ฅ2+ 2)/(๐‘ฅ2+ 3)(๐‘ฅ2+ 4) = โˆซ1โ–’ใ€–1โˆ’[(โˆ’2)/((๐‘ฅ^2 + 3) ) + 6/((๐‘ฅ^2 + 4) )] ใ€— ๐‘‘๐‘ฅ = โˆซ1โ–’1. ๐‘‘๐‘ฅ + โˆซ1โ–’2/((๐‘ฅ^2 + 3) ) ๐‘‘๐‘ฅ โˆ’ โˆซ1โ–’6/((๐‘ฅ^2 + 4) ) ๐‘‘๐‘ฅ = โˆซ1โ–’1. ๐‘‘๐‘ฅ + 2โˆซ1โ–’1/(๐‘ฅ^2 + (โˆš3)^2 ) ๐‘‘๐‘ฅ โˆ’ 6โˆซ1โ–’1/((๐‘ฅ^2 +2^2 ) ) ๐‘‘๐‘ฅ = ๐‘ฅ + 2 ร— 1/โˆš3 tan^(โˆ’1)โกใ€– ๐‘ฅ/โˆš3ใ€— โˆ’ 6 ร— 1/2 tan^(โˆ’1)โกใ€– ๐‘ฅ/2ใ€—+๐ถ = ๐’™ + ๐Ÿ/โˆš๐Ÿ‘ ใ€–๐’•๐’‚๐’ใ€—^(โˆ’๐Ÿ)โก(๐’™/โˆš๐Ÿ‘)โˆ’๐Ÿ‘ ใ€–๐’•๐’‚๐’ใ€—^(โˆ’๐Ÿ)โก(๐’™/๐Ÿ)+๐‘ช

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.