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

Transcript

Ex 7.11,7 By using the properties of definite integrals, evaluate the integrals : โˆซ_0^1โ–’ใ€– ๐‘ฅ(1โˆ’๐‘ฅ)^๐‘› ใ€— ๐‘‘๐‘ฅ Let I=โˆซ_0^1โ–’ใ€–๐‘ฅ(1โˆ’๐‘ฅ)^๐‘› ๐‘‘๐‘ฅใ€— โˆด I=โˆซ_0^1โ–’ใ€–(1โˆ’๐‘ฅ) [1โˆ’(1โˆ’๐‘ฅ)]^๐‘› ๐‘‘๐‘ฅใ€— I=โˆซ_0^1โ–’ใ€–(1โˆ’๐‘ฅ) [1โˆ’1+๐‘ฅ]^๐‘› ๐‘‘๐‘ฅใ€— I=โˆซ_0^1โ–’ใ€–(1โˆ’๐‘ฅ) [๐‘ฅ]^๐‘› ๐‘‘๐‘ฅใ€— I= โˆซ_0^1โ–’ใ€–(1โˆ’๐‘ฅ) ใ€– ๐‘ฅใ€—^๐‘› ๐‘‘๐‘ฅใ€— Using P4 : โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅ=ใ€— โˆซ_0^๐‘Žโ–’๐‘“(๐‘Žโˆ’๐‘ฅ)๐‘‘๐‘ฅ I= โˆซ_0^1โ–’ใ€–(ใ€– ๐‘ฅใ€—^๐‘›โˆ’ ๐‘ฅ^(๐‘› + 1) ) ๐‘‘๐‘ฅใ€— I= โˆซ_0^1โ–’ใ€–ใ€– ๐‘ฅใ€—^๐‘› ๐‘‘๐‘ฅใ€—โˆ’โˆซ_0^1โ–’ใ€–ใ€– ๐‘ฅใ€—^(๐‘› + 1) ๐‘‘๐‘ฅใ€— I=[๐‘ฅ^(๐‘› + 1)/(๐‘› + 1)]_0^1โˆ’[๐‘ฅ^(๐‘› + 2)/(๐‘› + 2)]_0^1 I=[(1)^(๐‘› + 1)/(๐‘› + 1)โˆ’(0)^(๐‘› + 1)/(๐‘› + 1)]โˆ’[(1)^(๐‘› + 2)/(๐‘› + 2)โˆ’(0)^(๐‘› + 2)/(๐‘› + 2)] I= 1/(๐‘› + 1)โˆ’1/(๐‘› + 2) I=(๐‘› + 2 โˆ’ (๐‘› + 1))/(๐‘› + 1)(๐‘› + 2) ๐ˆ=๐Ÿ/(๐’ + ๐Ÿ)(๐’ + 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.