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Ex 7.11, 5 - Using properties of definite integrals, |x + 2| dx - Definate Integration by properties - P2

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  1. Chapter 7 Class 12 Integrals
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Ex 7.11, 5 By using the properties of definite integrals, evaluate the integrals : โˆซ_(โˆ’5)^5โ–’ใ€– |๐‘ฅ+2| ใ€— ๐‘‘๐‘ฅ |๐‘ฅ+2|={โ–ˆ((๐‘ฅ+2) ๐‘–๐‘“ ๐‘ฅ+2โ‰ฅ0@โˆ’(๐‘ฅ+2) ๐‘–๐‘“ ๐‘ฅ+2<0)โ”ค ={โ–ˆ((๐‘ฅ+2) ๐‘–๐‘“ ๐‘ฅโ‰ฅโˆ’,2@โˆ’(๐‘ฅ+2) ๐‘–๐‘“ ๐‘ฅ<โˆ’2)โ”ค โˆด โˆซ_(โˆ’5)^5โ–’ใ€–|๐‘ฅ+2|๐‘‘๐‘ฅ=โˆซ_(โˆ’5)^(โˆ’2)โ–’ใ€–|๐‘ฅ+2|๐‘‘๐‘ฅ+ใ€—ใ€— โˆซ_(โˆ’2)^5โ–’|๐‘ฅ+2|๐‘‘๐‘ฅ =โˆซ_(โˆ’5)^(โˆ’2)โ–’ใ€–(๐‘ฅ+2)๐‘‘๐‘ฅ+ใ€— โˆซ_(โˆ’2)^5โ–’ใ€–+(๐‘ฅ+2)๐‘‘๐‘ฅใ€— =โˆซ_(โˆ’5)^(โˆ’2)โ–’ใ€–๐‘ฅ๐‘‘๐‘ฅโˆ’ใ€— โˆซ_(โˆ’5)^(โˆ’2)โ–’ใ€–2๐‘‘๐‘ฅโˆ’โˆซ_(โˆ’2)^5โ–’ใ€–๐‘ฅ๐‘‘๐‘ฅโˆ’โˆซ_(โˆ’2)^5โ–’2๐‘‘๐‘ฅใ€—ใ€— =[๐‘ฅ^2/2]_(โˆ’5)^(โˆ’2)+2[๐‘ฅ]_(โˆ’5)^(โˆ’2)โˆ’[๐‘ฅ^2/2]_(โˆ’2)^5โˆ’2[๐‘ฅ]_(โˆ’2)^5 =[๐‘ฅ^2/2]_(โˆ’5)^(โˆ’2)+2[๐‘ฅ]_(โˆ’5)^(โˆ’2)+[๐‘ฅ^2/2]_(โˆ’2)^5โˆ’2[๐‘ฅ]_(โˆ’2)^5 =(โˆ’(โˆ’2)^2+ (โˆ’5)^2)/2โˆ’2[โˆ’2โˆ’(โˆ’5)]+[((5)^2โˆ’(โˆ’2)^2)/2+2[5โˆ’(โˆ’2)]] =(4 + 25)/2โˆ’2[โˆ’2+5]+[(25 โˆ’ 4)/2]+2[5+2] =(+ 21)/2โˆ’2[3]+21/2+2[7] =(+ 21)/2+(โˆ’ 21)/2โˆ’6+14 =42/2+8 = 21+8 = ๐Ÿ๐Ÿ—

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