Misc 35 - Prove definite integral 0->1 x ex dx = 1 - Miscellaneous

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

Transcript

Misc 35 Prove that โˆซ_0^1โ–’๐‘ฅ ๐‘’^๐‘ฅ ๐‘‘๐‘ฅ=1 Solving L.H.S โˆซ_0^1โ–’๐‘ฅ ๐‘’^๐‘ฅ ๐‘‘๐‘ฅ First we will solve โˆซ1โ–’๐’™ ๐’†^๐’™ ๐’…๐’™ โˆซ1โ–’๐‘ฅ ๐‘’^๐‘ฅ ๐‘‘๐‘ฅ Using ILATE, we take First function :- ๐‘“(๐‘ฅ)=๐‘ฅ Second function :- g(๐‘ฅ)=๐‘’^๐‘ฅ โˆซ1โ–’๐‘ฅ ๐‘’^๐‘ฅ ๐‘‘๐‘ฅ=๐‘ฅโˆซ1โ–’ใ€–๐‘’^๐‘ฅ ๐‘‘๐‘ฅใ€—โˆ’โˆซ1โ–’(๐‘‘๐‘ฅ/๐‘‘๐‘ฅ โˆซ1โ–’ใ€–๐‘’^๐‘ฅ ๐‘‘๐‘ฅใ€—) ๐‘‘๐‘ฅ = ๐‘ฅ๐‘’^๐‘ฅโˆ’โˆซ1โ–’1. ๐‘’^๐‘ฅ ๐‘‘๐‘ฅ = ๐‘ฅ๐‘’^๐‘ฅโˆ’๐‘’^๐‘ฅ+๐ถ = ๐‘ฅ logโก๐‘ฅโˆ’๐‘ฅ+๐ถ Applying limits โˆซ1_0^1โ–’ใ€–๐‘ฅ ๐‘’^๐‘ฅ ๐‘‘๐‘ฅใ€— = [๐‘ฅ๐‘’^๐‘ฅโˆ’๐‘’^๐‘ฅ ]_0^1 = (1๐‘’^1โˆ’๐‘’^1 )โˆ’(0.๐‘’^0โˆ’๐‘’^0) = (๐‘’^1โˆ’๐‘’^1 )โˆ’ (โˆ’ 1) = 1 = R.H.S Hence, proved.

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Davneet Singh
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.