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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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