Ex 7.9, 15 - Direct Integrate x ex2 dx from 0 to 1 - Ex 7.9

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex7.9, 15 โˆซ_0^1โ–’ใ€–๐‘ฅ ใ€–๐‘’^๐‘ฅใ€—^2 ใ€— ๐‘‘๐‘ฅ Step 1 :- Let F(๐‘ฅ)=โˆซ1โ–’ใ€–๐‘ฅ ๐‘’^(๐‘ฅ^2 ) ๐‘‘๐‘ฅใ€— Let ๐‘ฅ^2=๐‘ก Differentiating w.r.t.๐‘ฅ ๐‘‘/๐‘‘๐‘ฅ (๐‘ฅ^2 )=๐‘‘๐‘ก/๐‘‘๐‘ฅ 2๐‘ฅ=๐‘‘๐‘ก/๐‘‘๐‘ฅ ๐‘‘๐‘ฅ=๐‘‘๐‘ก/2๐‘ฅ Therefore, โˆซ1โ–’ใ€–๐‘ฅ ๐‘’^(๐‘ฅ^2 ) ๐‘‘๐‘ฅ=โˆซ1โ–’ใ€–๐‘ฅ ๐‘’^๐‘ก ๐‘‘๐‘ก/2๐‘ฅใ€—ใ€— =1/2 โˆซ1โ–’ใ€–๐‘’^๐‘ก ๐‘‘๐‘กใ€— =1/2 ๐‘’^๐‘ก Putting ๐‘ก=๐‘ฅ^2 =1/2 ๐‘’^(๐‘ฅ^2 ) Hence F(๐‘ฅ)=1/2 ๐‘’^(๐‘ฅ^2 ) Step 2 :- โˆซ_0^1โ–’ใ€–๐‘ฅ๐‘’^๐‘ฅ ๐‘‘๐‘ฅใ€—=๐น(1)โˆ’๐น(0) =1/2 ๐‘’^(1^2 )โˆ’1/2 ๐‘’^(0^2 ) =1/2 ๐‘’^1โˆ’1/2 ๐‘’^0 =1/2 (๐‘’โˆ’1)

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.