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Ex 7.2, 20 - Integrate (e^2x - e^-2x) / (e^2x + e^-2x) - Ex 7.2

Ex 7.2, 20 - Chapter 7 Class 12 Integrals - Part 2

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Ex 7.2, 20 Integrate the function (๐‘’^2๐‘ฅ โˆ’ ๐‘’^(โˆ’2๐‘ฅ))/(๐‘’^2๐‘ฅ + ๐‘’^(โˆ’2๐‘ฅ) ) Let ๐‘’^2๐‘ฅ + ๐‘’^(โˆ’2๐‘ฅ)= ๐‘ก Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘’^2๐‘ฅ. ๐‘‘(2๐‘ฅ)/๐‘‘๐‘ฅ +๐‘’^(โˆ’2๐‘ฅ) ๐‘‘(โˆ’2๐‘ฅ)/๐‘‘๐‘ฅ= ๐‘‘๐‘ก/๐‘‘๐‘ฅ ใ€–2๐‘’ใ€—^2๐‘ฅโˆ’ใ€–2๐‘’ใ€—^(โˆ’2๐‘ฅ)= ๐‘‘๐‘ก/๐‘‘๐‘ฅ 2(๐‘’^2๐‘ฅโˆ’๐‘’^(โˆ’2๐‘ฅ) )=๐‘‘๐‘ก/๐‘‘๐‘ฅ " " ๐‘‘๐‘ฅ = ๐‘‘๐‘ก/2(๐‘’^2๐‘ฅโˆ’ ๐‘’^(โˆ’2๐‘ฅ) ) Integrating the function โˆซ1โ–’ใ€–" " (๐‘’^2๐‘ฅ โˆ’ ๐‘’^(โˆ’2๐‘ฅ))/(๐‘’^2๐‘ฅ + ๐‘’^(โˆ’2๐‘ฅ) )ใ€—. ๐‘‘๐‘ฅ Putting ๐‘’^2๐‘ฅ + ๐‘’^(โˆ’2๐‘ฅ)=๐‘ก & ๐‘‘๐‘ฅ=๐‘‘๐‘ก/2(๐‘’^2๐‘ฅโˆ’ ๐‘’^(โˆ’2๐‘ฅ) ) = โˆซ1โ–’ใ€–" " (๐‘’^2๐‘ฅ โˆ’ ๐‘’^(โˆ’2๐‘ฅ))/๐‘กใ€—. ๐‘‘๐‘ก/2(๐‘’^2๐‘ฅโˆ’ ๐‘’^(โˆ’2๐‘ฅ) ) = โˆซ1โ–’ใ€–" " 1/2๐‘กใ€—. ๐‘‘๐‘ก = 1/2 โˆซ1โ–’1/๐‘ก. ๐‘‘๐‘ก = 1/2 logโกใ€– |๐‘ก|ใ€—+๐ถ = 1/2 logโกใ€– |๐‘’^2๐‘ฅ + ๐‘’^(โˆ’2๐‘ฅ) |ใ€—+๐ถ = ๐Ÿ/๐Ÿ ๐’๐’๐’ˆโกใ€– (๐’†^๐Ÿ๐’™ + ๐’†^(โˆ’๐Ÿ๐’™) )ใ€—+๐‘ช (Using โˆซ1โ–’1/๐‘ฅ. ๐‘‘๐‘ฅ=๐‘™๐‘œ๐‘”โก|๐‘ฅ| ) (Using ๐‘ก=๐‘’^2๐‘ฅ + ๐‘’^(โˆ’2๐‘ฅ)) (โˆด ๐‘’^2๐‘ฅ+๐‘’^(โˆ’2๐‘ฅ)>0 )

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.