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Ex 7.5, 6 - Integrate 1 - x2 / x (1 - 2x) - Class 12 - Ex 7.5

Ex 7.5, 6 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.5, 6 - Chapter 7 Class 12 Integrals - Part 3 Ex 7.5, 6 - Chapter 7 Class 12 Integrals - Part 4 Ex 7.5, 6 - Chapter 7 Class 12 Integrals - Part 5

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Ex 7.5, 6 Integrate the function (1 โˆ’ ๐‘ฅ2)/(๐‘ฅ(1 โˆ’ 2๐‘ฅ)) โˆซ1โ–’(1 โˆ’ ๐‘ฅ2)/(๐‘ฅ(1 โˆ’ 2๐‘ฅ)) ๐‘‘๐‘ฅ= โˆซ1โ–’(1 โˆ’ ๐‘ฅ^2)/(๐‘ฅ โˆ’ 2๐‘ฅ^2 ) ๐‘‘๐‘ฅ =โˆซ1โ–’(1/2 + (โˆ’ ๐‘ฅ/2 + 1)/(โˆ’2๐‘ฅ^2+ ๐‘ฅ)) ๐‘‘๐‘ฅ =1/2 โˆซ1โ–’๐‘‘๐‘ฅ+1/2 โˆซ1โ–’(โˆ’๐‘ฅ + 2)/(โˆ’2๐‘ฅ^2+ ๐‘ฅ) ๐‘‘๐‘ฅ =1/2 โˆซ1โ–’๐‘‘๐‘ฅโˆ’1/2 โˆซ1โ–’(๐‘ฅ โˆ’ 2)/๐‘ฅ(1 โˆ’ 2๐‘ฅ) ๐‘‘๐‘ฅ =1/2 โˆซ1โ–’๐‘‘๐‘ฅโˆ’1/2 โˆซ1โ–’(๐‘ฅ โˆ’ 2)/๐‘ฅ(2๐‘ฅ โˆ’ 1) ๐‘‘๐‘ฅ โˆ’2๐‘ฅ^2+๐‘ฅ โˆ’๐‘ฅ^2+๐‘ฅ/2 Now Solving (๐‘ฅ โˆ’ 2)/(๐‘ฅ (2๐‘ฅ โˆ’ 1) ) = ๐ด/๐‘ฅ + ๐ต/(2๐‘ฅ โˆ’ 1) (๐‘ฅ โˆ’ 2)/(๐‘ฅ (2๐‘ฅ โˆ’ 1) ) = (๐ด(2๐‘ฅ โˆ’ 1) + ๐ต๐‘ฅ)/(๐‘ฅ (2๐‘ฅ โˆ’ 1) ) Cancelling denominator ๐‘ฅโˆ’2=๐ด(2๐‘ฅโˆ’1)+๐ต๐‘ฅ Putting x = 0 in (2) ๐‘ฅโˆ’2=๐ด(2๐‘ฅโˆ’1)+๐ต๐‘ฅ 0โˆ’2 = ๐ด(2ร—0โˆ’1) + ๐ตร—0 โˆ’2 = A(โˆ’1) โ€ฆ(2) โˆ’2 = โˆ’๐ด ๐ด = 2 Similarly Putting x = 1/2 in (2) ๐‘ฅโˆ’2=๐ด(2๐‘ฅโˆ’1)+๐ต๐‘ฅ 1/2 โˆ’ 2 = A(1/2ร—2โˆ’1)+๐ตร—1/2 (โˆ’3)/2 = A(1โˆ’1)+๐ตร—1/2 (โˆ’3)/2 = Aร—0+ ๐ต/2 (โˆ’3)/2 = ๐ต/2 ๐ต = โˆ’3 Hence we can write it as (๐‘ฅ โˆ’ 2)/(๐‘ฅ (2๐‘ฅ โˆ’ 1) ) = ๐ด/๐‘ฅ + ๐ต/(2๐‘ฅ โˆ’ 1) (๐‘ฅ โˆ’ 2)/(๐‘ฅ (2๐‘ฅ โˆ’ 1) ) = 2/๐‘ฅ + ((โˆ’3))/(2๐‘ฅ โˆ’ 1) = 2/๐‘ฅ โˆ’ 3/(2๐‘ฅ โˆ’ 1) Therefore , from (1) we get, โˆซ1โ–’(1 โˆ’ ๐‘ฅ^2)/๐‘ฅ(1 โˆ’ 2๐‘ฅ) =1/2 โˆซ1โ–’๐‘‘๐‘ฅ+ 1/2 โˆซ1โ–’(2/๐‘ฅ โˆ’ 3/(2๐‘ฅ โˆ’ 1)) ๐‘‘๐‘ฅ =1/2 โˆซ1โ–’๐‘‘๐‘ฅ+ 2/2 โˆซ1โ–’ใ€–๐‘‘๐‘ฅ/๐‘ฅ โˆ’3/2ใ€— โˆซ1โ–’๐‘‘๐‘ฅ/(2๐‘ฅ โˆ’ 1) =1/2 โˆซ1โ–’๐‘‘๐‘ฅ+โˆซ1โ–’ใ€–๐‘‘๐‘ฅ/๐‘ฅ + 3/2ใ€— โˆซ1โ–’๐‘‘๐‘ฅ/(1 โˆ’ 2๐‘ฅ) =1/2 ๐‘ฅ+logโก|๐‘ฅ|+3/2 logโก|1 โˆ’ 2๐‘ฅ|/(โˆ’2) +๐ถ =๐’™/๐Ÿ +๐’๐’๐’ˆโก|๐’™|โˆ’๐Ÿ‘/๐Ÿ’ ๐’๐’๐’ˆโก|๐Ÿโˆ’๐Ÿ๐’™|+๐‘ช

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.