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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise

Transcript

Misc 2 Integrate the function 1/(โˆš(๐‘ฅ + ๐‘Ž) + โˆš(๐‘ฅ +๐‘) ) โˆซ1โ–’ใ€–1/(โˆš(๐‘ฅ + ๐‘Ž) + โˆš(๐‘ฅ + ๐‘) ) ๐‘‘๐‘ฅใ€— Rationalising = โˆซ1โ–’ใ€–(1/(โˆš(๐‘ฅ + ๐‘Ž) + โˆš(๐‘ฅ + ๐‘) ) ร— (โˆš(๐‘ฅ + ๐‘Ž) โˆ’ โˆš(๐‘ฅ + ๐‘))/(โˆš(๐‘ฅ + ๐‘Ž) โˆ’ โˆš(๐‘ฅ + ๐‘) )) ๐‘‘๐‘ฅใ€— = โˆซ1โ–’ใ€–(โˆš(๐‘ฅ + ๐‘Ž) โˆ’ โˆš(๐‘ฅ + ๐‘))/((โˆš(๐‘ฅ + ๐‘Ž) + โˆš(๐‘ฅ + ๐‘) )(โˆš(๐‘ฅ + ๐‘Ž) โˆ’ โˆš(๐‘ฅ + ๐‘) ) ) ๐‘‘๐‘ฅใ€— Using (๐‘Žโˆ’๐‘)(๐‘Ž+๐‘)=๐‘Ž^2โˆ’๐‘^2 =โˆซ1โ–’ใ€–(โˆš(๐‘ฅ + ๐‘Ž) โˆ’ โˆš(๐‘ฅ + ๐‘))/((โˆš(๐‘ฅ + ๐‘Ž) )^2 โˆ’ (โˆš(๐‘ฅ + ๐‘) )^2 ) ๐‘‘๐‘ฅใ€— = โˆซ1โ–’ใ€–(โˆš(๐‘ฅ + ๐‘Ž) โˆ’ โˆš(๐‘ฅ + ๐‘))/(๐‘ฅ + ๐‘Ž โˆ’(๐‘ฅ + ๐‘) ) ๐‘‘๐‘ฅใ€— = โˆซ1โ–’ใ€–(โˆš(๐‘ฅ + ๐‘Ž) โˆ’ โˆš(๐‘ฅ + ๐‘))/(๐‘Ž โˆ’ ๐‘) ๐‘‘๐‘ฅใ€— = 1/(๐‘Ž โˆ’ ๐‘) โˆซ1โ–’(โˆš(๐‘ฅ+๐‘Ž) โˆ’โˆš(๐‘ฅ+๐‘) ) ๐‘‘๐‘ฅ = 1/(๐‘Ž โˆ’ ๐‘) โˆซ1โ–’((๐‘ฅ+๐‘Ž)^(1/2) โˆ’(๐‘ฅ+๐‘Ž)^(1/2) ) ๐‘‘๐‘ฅ = 1/(๐‘Ž โˆ’ ๐‘) [โˆซ1โ–’(๐‘ฅ+๐‘Ž)^(1/2) ๐‘‘๐‘ฅโˆ’โˆซ1โ–’(๐‘ฅ+๐‘Ž)^(1/2) ๐‘‘๐‘ฅ] = 1/(๐‘Ž โˆ’ ๐‘) [(๐‘ฅ + ๐‘Ž)^(1/2 + 1)/(1/2 + 1) โˆ’ (๐‘ฅ + ๐‘)^(1/2 + 1)/(1/2 + 1) ] + ๐ถ = 1/(๐‘Ž โˆ’ ๐‘) [(๐‘ฅ + ๐‘Ž)^(3/2)/(3/2) โˆ’ (๐‘ฅ + ๐‘)^(3/2)/(3/2)] + ๐ถ = 1/(๐‘Ž โˆ’ ๐‘) [ใ€–2(๐‘ฅ + ๐‘Ž)ใ€—^(3/2)/3 โˆ’ ใ€–2(๐‘ฅ + ๐‘)ใ€—^(3/2)/3] + ๐ถ = ๐Ÿ/๐Ÿ‘(๐’‚ โˆ’ ๐’ƒ) [(๐’™+๐’‚)^(๐Ÿ‘/๐Ÿ)โˆ’(๐’™+๐’ƒ)^(๐Ÿ‘/๐Ÿ) ] + ๐‘ช

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.