Ex 7.11, 19 - Show f(x) g(x) dx = 2 f(x), f(x) = f(a-x) - Definate Integration by properties - P4

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.11, 19 Show that โˆซ_0^๐‘Žโ–’๐‘“(๐‘ฅ) ๐‘” (๐‘ฅ) ๐‘‘๐‘ฅ=2โˆซ_0^๐‘Žโ–’๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ, if f and g are defined as ๐‘“(๐‘ฅ)=๐‘“(๐‘Žโˆ’๐‘ฅ) and ๐‘”(๐‘ฅ)+๐‘”(๐‘Žโˆ’๐‘ฅ)=4 Let I =โˆซ_0^๐‘Žโ–’๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) ๐‘‘๐‘ฅ I =โˆซ_0^๐‘Žโ–’๐‘“(๐‘ฅ) [4โˆ’๐‘”(๐‘Žโˆ’๐‘ฅ)] ๐‘‘๐‘ฅ I = โˆซ_0^๐‘Žโ–’[4.๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅ)๐‘”(๐‘Žโˆ’๐‘ฅ)] ๐‘‘๐‘ฅ I = 4โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅโˆ’โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ) ๐‘”(๐‘Žโˆ’๐‘ฅ) ใ€—ใ€— ๐‘‘๐‘ฅ I = 4โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅโˆ’โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘Žโˆ’๐‘ฅ) ๐‘”(๐‘Žโˆ’(๐‘Žโˆ’๐‘ฅ)) ใ€—ใ€— ๐‘‘๐‘ฅ I = 4โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅโˆ’โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) ใ€—ใ€— ๐‘‘๐‘ฅ I =4โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅโˆ’Iใ€— I +I=4โˆซ_0^๐‘Žโ–’๐‘“(๐‘ฅ)๐‘‘๐‘ฅ 2I=4โˆซ_0^๐‘Žโ–’๐‘“(๐‘ฅ)๐‘‘๐‘ฅ I=2โˆซ_0^๐‘Žโ–’๐‘“(๐‘ฅ)๐‘‘๐‘ฅ โˆด โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ) ๐‘”(๐‘ฅ) ใ€— ๐‘‘๐‘ฅ=2โˆซ_0^๐‘Žโ–’๐‘“(๐‘ฅ)๐‘‘๐‘ฅ Hence Proved

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