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

Transcript

Ex 7.11, 14 By using the properties of definite integrals, evaluate the integrals : โˆซ_0^2๐œ‹โ–’cos^5โก๐‘ฅ ๐‘‘๐‘ฅ โˆซ_0^2๐œ‹โ–’cos^5โก๐‘ฅ ๐‘‘๐‘ฅ =โˆซ_0^๐œ‹โ–’cos^5โก๐‘ฅ ๐‘‘๐‘ฅ+โˆซ_0^๐œ‹โ–’ใ€–cos^5 (2ฯ€โˆ’๐‘ฅ)ใ€— ๐‘‘๐‘ฅ = โˆซ_0^๐œ‹โ–’ใ€–ใ€–๐‘๐‘œ๐‘ ใ€—^5 ๐‘ฅ ๐‘‘๐‘ฅ+โˆซ_0^๐œ‹โ–’ใ€–๐‘๐‘œ๐‘ ใ€—^5 ใ€— ๐‘ฅ = 2 โˆซ_0^๐œ‹โ–’ใ€–ใ€–๐‘๐‘œ๐‘ ใ€—^5 ๐‘ฅ ๐‘‘๐‘ฅใ€— Using property: โˆซ_0^2๐‘Žโ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅ=โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅ+โˆซ_0^๐‘Žโ–’๐‘“(2๐‘Žโˆ’๐‘ฅ)๐‘‘๐‘ฅใ€—ใ€— (As cos (2ฯ€ โˆ’ ๐œƒ) = cos ๐œƒ) Using property: โˆซ_0^2๐‘Žโ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅ=โˆซ_0^๐‘Žโ–’ใ€–๐‘“(๐‘ฅ)๐‘‘๐‘ฅ+โˆซ_0^๐‘Žโ–’๐‘“(2๐‘Žโˆ’๐‘ฅ)๐‘‘๐‘ฅใ€—ใ€— = 2 (โˆซ_0^(๐œ‹/2)โ–’ใ€–ใ€–๐‘๐‘œ๐‘ ใ€—^5 ๐‘ฅ ๐‘‘๐‘ฅ+โˆซ_0^(๐œ‹/2)โ–’ใ€–cosโก(๐œ‹โˆ’๐‘ฅ) ใ€—ใ€— ๐‘‘๐‘ฅ) = 2 (โˆซ_0^(๐œ‹/2)โ–’ใ€–ใ€–๐‘๐‘œ๐‘ ใ€—^5 ๐‘ฅ ๐‘‘๐‘ฅ+โˆซ_0^(๐œ‹/2)โ–’ใ€–(โˆ’ cos ๐‘ฅ)^5โก๐‘‘๐‘ฅ ใ€—ใ€—) = 2 (โˆซ_0^(๐œ‹/2)โ–’ใ€–ใ€–๐‘๐‘œ๐‘ ใ€—^5 ๐‘ฅ ๐‘‘๐‘ฅโˆ’โˆซ_0^(๐œ‹/2)โ–’ใ€–ใ€–ใ€–๐‘๐‘œ๐‘ ใ€—^5 ๐‘ฅใ€—โก๐‘‘๐‘ฅ ใ€—ใ€—) = 2ร—0 = 0 (cos (๐œ‹โˆ’๐œƒ) = โ€“ cos ๐œƒ)

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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.