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

Transcript

Ex 7.2, 19 Integrate the function (๐‘’2๐‘ฅ โˆ’ 1)/(๐‘’2๐‘ฅ+ 1) Simplify the given function (๐‘’^2๐‘ฅ โˆ’ 1)/(๐‘’^2๐‘ฅ + 1) Dividing numerator and denominator by ex, we obtain = (๐‘’^2๐‘ฅ/๐‘’^๐‘ฅ " " โˆ’" " ๐Ÿ/๐’†^๐’™ )/(๐‘’^2๐‘ฅ/๐‘’^๐‘ฅ " " + " " ๐Ÿ/๐’†^๐’™ ) = (๐‘’^๐’™ โˆ’ ๐’†^(โˆ’๐’™))/(๐‘’^๐’™ + ๐’†^(โˆ’๐’™) ) Let ๐‘’^๐‘ฅ + ๐‘’^(โˆ’๐‘ฅ)= ๐‘ก Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘’^๐‘ฅ+(โˆ’1) ๐‘’^(โˆ’๐‘ฅ)= ๐‘‘๐‘ก/๐‘‘๐‘ฅ ๐‘’^๐‘ฅโˆ’๐‘’^(โˆ’๐‘ฅ)= ๐‘‘๐‘ก/๐‘‘๐‘ฅ ๐‘‘๐‘ฅ=๐‘‘๐‘ก/(๐‘’^๐‘ฅ โˆ’ ๐‘’^(โˆ’๐‘ฅ) ) Now, Integrating the function โˆซ1โ–’ใ€–" " (๐‘’^2๐‘ฅ โˆ’ 1)/(๐‘’^2๐‘ฅ + 1) " " ใ€—. ๐‘‘๐‘ฅ = โˆซ1โ–’ใ€–" " (๐‘’^๐‘ฅ โˆ’ ๐‘’^(โˆ’๐‘ฅ))/(๐‘’^๐‘ฅ + ๐‘’^(โˆ’๐‘ฅ) ) " " ใ€—. ๐‘‘๐‘ฅ (Using (๐‘’^2๐‘ฅ โˆ’ 1)/(๐‘’^2๐‘ฅ + 1)=(๐‘’^๐‘ฅ โˆ’ ๐‘’^(โˆ’๐‘ฅ))/(๐‘’^๐‘ฅ + ๐‘’^(โˆ’๐‘ฅ) ) ) Putting ๐‘’^๐‘ฅ + ๐‘’^(โˆ’๐‘ฅ)=๐‘ก & ๐‘‘๐‘ฅ=๐‘‘๐‘ก/(๐‘’^๐‘ฅ โˆ’ ๐‘’^(โˆ’๐‘ฅ) ) =โˆซ1โ–’ใ€–" " (๐‘’^๐‘ฅ โˆ’ ๐‘’^(โˆ’๐‘ฅ))/๐‘ก " " ใ€—. ๐‘‘๐‘ก/(๐‘’^๐‘ฅ โˆ’ ๐‘’^(โˆ’๐‘ฅ) ) " " =โˆซ1โ–’ใ€–" " 1/๐‘ก " " ใ€—. ๐‘‘๐‘ก =logโก|๐‘ก|+๐ถ =logโกใ€– |๐‘’^๐‘ฅ+๐‘’^(โˆ’๐‘ฅ) |ใ€—+๐ถ =๐’๐’๐’ˆโก(๐’†^๐’™ + ๐’†^(โˆ’๐’™) )+๐‘ช (Using ๐‘ก=๐‘’^๐‘ฅ + ๐‘’^(โˆ’๐‘ฅ)) (As ๐‘’^๐‘ฅ+๐‘’^(โˆ’๐‘ฅ)>0 )

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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.