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Ex 7.5, 21 - Integrate 1 / (ex - 1) [Hing: Put ex = t] - Integration by partial fraction - Type 1

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.5, 21 1﷮( 𝑒﷮𝑥﷯ − 1) ﷯ [Hint : Put ex = t] Let 𝑒﷮𝑥﷯ = 𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑒﷮𝑥﷯ = 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥 = 𝑑𝑡﷮ 𝑒﷮𝑥﷯﷯ Therefore ﷮﷮ 1﷮( 𝑒﷮𝑥﷯ − 1) ﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ 𝑡 − 1﷯ ﷯﷯ 𝑑𝑡﷮ 𝑒﷮𝑥﷯﷯ We can write integrand as 1﷮𝑡 𝑡 − 1﷯ ﷯ = 𝐴﷮𝑡﷯ + 𝐵﷮𝑡 − 1﷯ 1﷮𝑡 𝑡 − 1﷯ ﷯ = 𝐴 𝑡 − 1﷯ + 𝐵 𝑡﷮𝑡 𝑡 − 1﷯﷯ By cancelling denominator 1 = 𝐴 𝑡−1﷯+𝐵 𝑡 Putting t = 0 in (1) 1 = 𝐴 0−1﷯+𝐵×0 1 = 𝐴× −1﷯ 1 = −𝐴 𝐴 = −1 Similarly putting t = 1 1 = 𝐴 1−1﷯+𝐵×1 1 = 𝐴×0+𝐵 1 = 𝐵 𝐵 = 1 Therefore ﷮﷮ 1﷮𝑡 𝑡 − 1﷯ ﷯﷯ 𝑑𝑡 = ﷮﷮ −1﷮𝑡 ﷯﷯ 𝑑𝑡 + ﷮﷮ 1﷮𝑡 − 1 ﷯﷯ = − log ﷮ 𝑡﷯﷯+ log ﷮ 𝑡−1﷯﷯+𝐶 = log ﷮ 𝑡 − 1﷮𝑡﷯﷯﷯+𝐶 Putting t = 𝑒﷮𝑥﷯ ﷮﷮ 1﷮ 𝑒﷮𝑥﷯ − 1﷯ ﷯﷯ 𝑑𝑥 = 𝑙𝑜𝑔 ﷮ 𝑒﷮𝑥﷯ − 1﷮ 𝑒﷮𝑥﷯﷯﷯﷯+𝐶 = 𝐥𝐨𝐠 ﷮ 𝒆﷮𝒙﷯ − 𝟏﷮ 𝒆﷮𝒙﷯﷯﷯﷯+𝑪

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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