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

Transcript

Ex 7.6, 3 Integrate the function ๐‘ฅ^2 ๐‘’๐‘ฅ โˆซ1โ–’ใ€–๐‘ฅ^2 ๐‘’^๐‘ฅ ๐‘‘๐‘ฅใ€— = ๐‘ฅ^2 โˆซ1โ–’ใ€–๐‘’๐‘ฅ ๐‘‘๐‘ฅใ€—โˆ’โˆซ1โ–’(๐‘‘(๐‘ฅ^2 )/๐‘‘๐‘ฅ โˆซ1โ–’ใ€–๐‘’๐‘ฅ ๐‘‘๐‘ฅใ€—) ๐‘‘๐‘ฅ = ๐‘ฅ^2. ๐‘’๐‘ฅ โˆ’โˆซ1โ–’ใ€–2๐‘ฅ . ๐‘’๐‘ฅใ€— ๐‘‘๐‘ฅ = ๐‘ฅ^2. ๐‘’๐‘ฅ โˆ’2โˆซ1โ–’ใ€–๐’™ . ๐’†๐’™ใ€— ๐’…๐’™ Now we know that โˆซ1โ–’ใ€–๐‘“(๐‘ฅ) ๐‘”โก(๐‘ฅ) ใ€— ๐‘‘๐‘ฅ=๐‘“(๐‘ฅ) โˆซ1โ–’๐‘”(๐‘ฅ) ๐‘‘๐‘ฅโˆ’โˆซ1โ–’(๐‘“โ€ฒ(๐‘ฅ)โˆซ1โ–’๐‘”(๐‘ฅ) ๐‘‘๐‘ฅ) ๐‘‘๐‘ฅ Putting f(x) = x2 and g(x) = ex โ€ฆ(1) Solving I1 โˆซ1โ–’ใ€–๐‘ฅ ๐‘’^๐‘ฅ ๐‘‘๐‘ฅใ€— = ๐‘ฅโˆซ1โ–’๐‘’๐‘ฅ ๐‘‘๐‘ฅโˆ’โˆซ1โ–’(๐‘‘๐‘ฅ/๐‘‘๐‘ฅ โˆซ1โ–’๐‘’^๐‘ฅ ๐‘‘๐‘ฅ) ๐‘‘๐‘ฅ = ๐‘ฅ๐‘’๐‘ฅ โˆ’โˆซ1โ–’๐‘’๐‘ฅ ๐‘‘๐‘ฅ = ๐‘ฅ๐‘’๐‘ฅ โˆ’๐‘’๐‘ฅ Now we know that โˆซ1โ–’ใ€–๐‘“(๐‘ฅ) ๐‘”โก(๐‘ฅ) ใ€— ๐‘‘๐‘ฅ=๐‘“(๐‘ฅ) โˆซ1โ–’๐‘”(๐‘ฅ) ๐‘‘๐‘ฅโˆ’โˆซ1โ–’(๐‘“โ€ฒ(๐‘ฅ)โˆซ1โ–’๐‘”(๐‘ฅ) ๐‘‘๐‘ฅ) ๐‘‘๐‘ฅ Putting f(x) = x and g(x) = ex Putting value of I1 in our equation โˆด โˆซ1โ–’ใ€–๐‘ฅ^2 ๐‘’๐‘ฅ" " ใ€— ๐‘‘๐‘ฅ" = " ๐‘ฅ^2. ๐‘’๐‘ฅ โˆ’2โˆซ1โ–’ใ€–๐’™ . ๐’†๐’™ใ€— ๐’…๐’™ =๐‘ฅ^2. ๐‘’๐‘ฅ โˆ’2(๐’™๐’†๐’™โˆ’๐’†^๐’™ )+๐ถ =๐‘ฅ^2. ๐‘’๐‘ฅ โˆ’2๐‘ฅ๐‘’๐‘ฅ+ใ€–2๐‘’ใ€—^๐‘ฅ+๐ถ =๐’†๐’™ (๐’™^๐Ÿโˆ’๐Ÿ๐’™+๐Ÿ)+๐‘ช

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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.