Integration Full Chapter Explained -



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


Ex 7.10, 8 Evaluate the integrals using substitution โˆซ_1^(2 )โ–’ใ€– (1/๐‘ฅ โˆ’1/(2๐‘ฅ^2 )) ใ€— ๐‘’^2๐‘ฅ ๐‘‘๐‘ฅ Let ๐‘ก=2๐‘ฅ ๐‘‘๐‘ก/๐‘‘๐‘ฅ=2 ๐‘‘๐‘ก/2=๐‘‘๐‘ฅ Thus, when x varies from 1 to 2, t varies from 2 to 4 Substituting, โˆซ_1^(2 )โ–’ใ€– (1/๐‘ฅ โˆ’1/(2๐‘ฅ^2 )) ใ€— ๐‘’^2๐‘ฅ ๐‘‘๐‘ฅ = โˆซ_2^4โ–’ใ€–๐‘’^๐‘ก (1/(๐‘ก/2)โˆ’1/(2ใ€– (๐‘ก/2)ใ€—^2 )) ใ€— ๐‘‘๐‘ก/2 =โˆซ_2^4โ–’ใ€–๐‘’^๐‘ก (2/๐‘กโˆ’4/(2๐‘ก^2 )) ใ€— ๐‘‘๐‘ก/2 =โˆซ_2^4โ–’ใ€–๐‘’^๐‘ก (1/๐‘กโˆ’2/๐‘ก^2 ) ใ€— ๐‘‘๐‘ก It is of the form โˆซ1โ–’ใ€–๐‘’^๐‘ฅ [๐‘“(๐‘ฅ)+๐‘“^โ€ฒ (๐‘ฅ)] ใ€— ๐‘‘๐‘ฅ=๐‘’^๐‘ฅ ๐‘“(๐‘ฅ)+๐ถ Where ๐‘“(๐‘ฅ)=1/๐‘ก ๐‘“^โ€ฒ (๐‘ฅ)= (โˆ’1)/๐‘ก^2 Hence, our equation becomes โˆซ_2^4โ–’ใ€–๐‘’^๐‘ก (1/๐‘กโˆ’2/๐‘ก^2 ) ใ€— ๐‘‘๐‘ก = [๐‘’^๐‘กร—1/๐‘ก]_2^4 = (๐‘’^4/4โˆ’๐‘’^2/2) = (๐’†^๐Ÿ (๐’†^๐Ÿ โˆ’ ๐Ÿ))/๐Ÿ’

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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.