Integration Full Chapter Explained - https://you.tube/Integration-Class-12

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

Transcript

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 Using the Property P2 โˆซ_๐‘Ž^๐‘โ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅ=โˆซ_๐‘Ž^๐‘โ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅ+โˆซ_๐‘^๐‘โ–’๐‘“(๐‘ฅ)๐‘‘๐‘ฅใ€—ใ€— =โˆซ_(โˆ’5)^(โˆ’2)โ–’ใ€–โˆ’(๐‘ฅ+2)๐‘‘๐‘ฅ+ใ€— โˆซ_(โˆ’2)^5โ–’(๐‘ฅ+2)๐‘‘๐‘ฅ =โˆ’โˆซ_(โˆ’5)^(โˆ’2)โ–’ใ€–๐‘ฅ๐‘‘๐‘ฅโˆ’ใ€— โˆซ_(โˆ’5)^(โˆ’2)โ–’2๐‘‘๐‘ฅ+โˆซ_(โˆ’2)^5โ–’๐‘ฅ๐‘‘๐‘ฅ+โˆซ_(โˆ’2)^5โ–’2๐‘‘๐‘ฅ =โˆ’[๐‘ฅ^2/2]_(โˆ’5)^(โˆ’2)โˆ’2[๐‘ฅ]_(โˆ’5)^(โˆ’2)+[๐‘ฅ^2/2]_(โˆ’2)^5+2[๐‘ฅ]_(โˆ’2)^5 =โˆ’(((โˆ’2)^2 โˆ’ (โˆ’5)^2)/2)โˆ’2[โˆ’2โˆ’(โˆ’5)]+[((5)^2 โˆ’ (โˆ’2)^2)/2] +2 [5โˆ’(โˆ’2)] =โˆ’((4 โˆ’ 25)/2)โˆ’2[โˆ’2+5]+[(25 โˆ’ 4)/2]+2[5+2] =โˆ’((โˆ’21)/2)โˆ’2[3]+21/2+2[7] =21/2+21/2โˆ’6+14 =42/2+8 = 21+8 = ๐Ÿ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.