Misc 3 fisrt ppt.jpg

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

Transcript

Misc 3 Integrate the function 1/(๐‘ฅ โˆš(๐‘Ž๐‘ฅ โˆ’ ๐‘ฅ^2 ) ) โˆซ1โ–’1/(๐‘ฅ โˆš(๐‘Ž๐‘ฅ โˆ’ ๐‘ฅ^2 ) ) ๐‘‘๐‘ฅ = โˆซ1โ–’1/(๐‘ฅ โˆš(๐‘ฅ^2 (๐‘Ž/๐‘ฅ โˆ’ 1) ) ) ๐‘‘๐‘ฅ = โˆซ1โ–’1/(๐‘ฅ . ๐‘ฅโˆš((๐‘Ž/๐‘ฅ โˆ’ 1) ) ) ๐‘‘๐‘ฅ = โˆซ1โ–’1/(๐‘ฅ^2 โˆš((๐‘Ž/๐‘ฅ โˆ’ 1) ) ) ๐‘‘๐‘ฅ Let ๐‘Ž/๐‘ฅ โˆ’1=๐‘ก Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ (โˆ’ ๐‘Ž)/๐‘ฅ^2 โˆ’0 = ๐‘‘๐‘ก/๐‘‘๐‘ฅ (โˆ’ ๐‘Ž)/๐‘ฅ^2 = ๐‘‘๐‘ก/๐‘‘๐‘ฅ ๐‘‘๐‘ฅ = (โˆ’ ๐‘ฅ^2)/๐‘Ž . ๐‘‘๐‘ก Putting the values of (a/xโˆ’1) and dx, we get โˆซ1โ–’1/(๐‘ฅ^2 โˆš((๐‘Ž/๐‘ฅ โˆ’ 1) ) ) ๐‘‘๐‘ฅ = โˆซ1โ–’1/(๐‘ฅ^2 โˆš๐‘ก ) ๐‘‘๐‘ฅ = โˆซ1โ–’1/(๐‘ฅ^2 โˆš๐‘ก )ร—(โˆ’ ๐‘ฅ^2)/๐‘Ž . ๐‘‘๐‘ก = โˆซ1โ–’(โˆ’1)/๐‘Ž ๐‘‘๐‘ก/โˆš๐‘ก = (โˆ’1)/๐‘Ž โˆซ1โ–’(๐‘ก)^(โˆ’ 1/2) ๐‘‘๐‘ก = (โˆ’1)/๐‘Ž [๐‘ก^(โˆ’ 1/2 + 1)/(โˆ’ 1/2 + 1)]+๐ถ = (โˆ’1)/๐‘Ž [๐‘ก^(1/2)/(1/2)] + ๐ถ = (โˆ’2)/๐‘Ž [โˆš๐‘ก] + ๐ถ Putting back t = ๐‘Ž/๐‘ฅโˆ’1 = (โˆ’2)/๐‘Ž โˆš(๐‘Ž/๐‘ฅ โˆ’1) + ๐ถ = (โˆ’๐Ÿ)/๐’‚ โˆš((๐’‚ โˆ’ ๐’™)/๐’™) + 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 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.