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

Transcript

Ex 7.4, 15 Integrate the function 1/โˆš((๐‘ฅ โˆ’ ๐‘Ž)(๐‘ฅ โˆ’ ๐‘)) โˆซ1โ–’1/โˆš((๐‘ฅ โˆ’ ๐‘Ž) (๐‘ฅ โˆ’ ๐‘)) ๐‘‘๐‘ฅ =โˆซ1โ–’1/โˆš(๐‘ฅ(๐‘ฅ โˆ’ ๐‘Ž) โˆ’ ๐‘Ž(๐‘ฅ โˆ’ ๐‘)) ๐‘‘๐‘ฅ =โˆซ1โ–’1/โˆš(๐‘ฅ^2 โˆ’ ๐‘๐‘ฅ โˆ’ ๐‘Ž๐‘ฅ + ๐‘Ž๐‘) ๐‘‘๐‘ฅ =โˆซ1โ–’1/โˆš(๐‘ฅ^2 โˆ’ ๐‘ฅ(๐‘Ž + ๐‘) + ๐‘Ž๐‘) ๐‘‘๐‘ฅ =โˆซ1โ–’1/โˆš(๐‘ฅ^2 โˆ’ 2(๐‘ฅ)((๐‘Ž + ๐‘)/2) + ๐‘Ž๐‘) ๐‘‘๐‘ฅ =โˆซ1โ–’1/โˆš(๐‘ฅ^2 โˆ’ 2(๐‘ฅ)((๐‘Ž + ๐‘)/2) + ((๐‘Ž + ๐‘)/2)^2โˆ’ ((๐‘Ž + ๐‘)/2)^2+ ๐‘Ž๐‘) ๐‘‘๐‘ฅ =โˆซ1โ–’1/โˆš((๐‘ฅ โˆ’ (๐‘Ž + ๐‘)/2)^2 โˆ’ ((๐‘Ž + ๐‘)/2)^2+ ๐‘Ž๐‘) ๐‘‘๐‘ฅ =โˆซ1โ–’1/โˆš((๐‘ฅ โˆ’ (๐‘Ž + ๐‘)/2)^2 โˆ’ ((๐‘Ž^2 + ๐‘^2+ 2๐‘Ž๐‘)/4) + ๐‘Ž๐‘) ๐‘‘๐‘ฅ =โˆซ1โ–’1/โˆš((๐‘ฅ โˆ’ (๐‘Ž + ๐‘)/2)^2+ (โˆ’ ๐‘Ž^2 โˆ’ ๐‘^2 โˆ’ 2๐‘Ž๐‘ + 4๐‘Ž๐‘)/4) ๐‘‘๐‘ฅ =โˆซ1โ–’1/โˆš((๐‘ฅ โˆ’ (๐‘Ž + ๐‘)/2)^2 + (โˆ’ ๐‘Ž^2 โˆ’ ๐‘^2 + 2๐‘Ž๐‘)/4) ๐‘‘๐‘ฅ =โˆซ1โ–’1/(โˆš((๐‘ฅ โˆ’ (๐‘Ž + ๐‘)/2)^2 โˆ’ ((๐‘Ž^2 + ๐‘^2 โˆ’ 2๐‘Ž๐‘)/4) ) ) ๐‘‘๐‘ฅ =โˆซ1โ–’1/(โˆš((๐‘ฅ โˆ’ (๐‘Ž + ๐‘)/2)^2 โˆ’ ((๐‘Ž โˆ’ ๐‘)/2)^2 ) ) ๐‘‘๐‘ฅ =๐‘™๐‘œ๐‘”โก|๐‘ฅ โˆ’ (๐‘Ž + ๐‘)/2 +โˆš((๐‘ฅ โˆ’ (๐‘Ž + ๐‘)/2)^2โˆ’ ((๐‘Ž โˆ’ ๐‘)/2)^2 )|+๐ถ =๐‘™๐‘œ๐‘”โก|๐‘ฅโˆ’ (๐‘Ž + ๐‘)/2 +โˆš(๐‘ฅ^2+((๐‘Ž + ๐‘)/2)^2โˆ’2(๐‘ฅ)((๐‘Ž + ๐‘)/2)โˆ’((๐‘Ž โˆ’ ๐‘)/2)^2 )|+๐ถ It is of form โˆซ1โ–’๐‘‘๐‘ฅ/โˆš(๐‘ฅ^2 โˆ’ ๐‘Ž^2 ) =๐‘™๐‘œ๐‘”โก|๐‘ฅ+โˆš(๐‘ฅ^2โˆ’๐‘Ž^2 )|+๐ถ โˆด Replacing ๐‘ฅ by (๐‘ฅโˆ’ (๐‘Ž + ๐‘)/2) and a by ((๐‘Ž โˆ’ ๐‘)/2) , we get =๐‘™๐‘œ๐‘”โก|๐‘ฅโˆ’ (๐‘Ž + ๐‘)/2 +โˆš(๐‘ฅ^2โˆ’2(๐‘ฅ)((๐‘Ž + ๐‘)/2)+((๐‘Ž + ๐‘)/2)^2โˆ’((๐‘Ž โˆ’ ๐‘)/2)^2 )|+๐ถ =๐‘™๐‘œ๐‘”โก|๐‘ฅโˆ’ (๐‘Ž + ๐‘)/2 +โˆš(๐‘ฅ^2โˆ’๐‘ฅ(๐‘Ž+๐‘)+(๐‘Ž^2 + ๐‘^2 + 2๐‘Ž๐‘)/4โˆ’(๐‘Ž^2 + ๐‘^2 โˆ’ 2๐‘Ž๐‘)/4)|+๐ถ =๐‘™๐‘œ๐‘”โก|๐‘ฅโˆ’ (๐‘Ž + ๐‘)/2 +โˆš(๐‘ฅ^2โˆ’๐‘ฅ(๐‘Ž+๐‘)+2๐‘Ž๐‘/4+2๐‘Ž๐‘/4) |+๐ถ =๐‘™๐‘œ๐‘”โก|๐‘ฅโˆ’ (๐‘Ž + ๐‘)/2 +โˆš(๐‘ฅ^2โˆ’๐‘ฅ(๐‘Ž+๐‘)+4๐‘Ž๐‘/4) |+๐ถ =๐‘™๐‘œ๐‘”โก|๐‘ฅโˆ’ (๐‘Ž + ๐‘)/2 +โˆš(๐‘ฅ^2โˆ’๐‘ฅ(๐‘Ž+๐‘)+๐‘Ž๐‘) |+๐ถ =๐‘™๐‘œ๐‘”โก|๐‘ฅโˆ’ (๐‘Ž + ๐‘)/2 +โˆš(๐‘ฅ^2โˆ’๐‘Ž๐‘ฅโˆ’๐‘๐‘ฅ+๐‘Ž๐‘) |+๐ถ =๐‘™๐‘œ๐‘”โก|๐‘ฅโˆ’ (๐‘Ž + ๐‘)/2 +โˆš(๐‘ฅ(๐‘ฅโˆ’๐‘Ž)โˆ’๐‘(๐‘ฅโˆ’๐‘Ž) ) |+๐ถ =๐’๐’๐’ˆโก|๐’™โˆ’ (๐’‚ + ๐’ƒ)/๐Ÿ +โˆš((๐’™โˆ’๐’‚)(๐’™โˆ’๐’ƒ) ) |+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.