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

Transcript

Ex 7.6, 14 - Chapter 7 Class 12 Integrals - Maths We have to find Integration of x (log x)^2 We do this using by parts x (log x)^2 Here, x is algebraic and log x is logrithmic So, log x is first function and x is second function Ex 7.6, 14 ใ€–๐‘ฅ(logโก๐‘ฅ)ใ€—^2 โˆซ1โ–’ใ€–๐‘ฅ(logโก๐‘ฅ )^2.๐‘‘๐‘ฅ " " ใ€— 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) = (log x)2 and g(x) = x โˆด โˆซ1โ–’ใ€–๐‘ฅ(logโก๐‘ฅ )^2.๐‘‘๐‘ฅใ€—=โˆซ1โ–’ใ€–(logโก๐‘ฅ )^2 ๐‘ฅ .๐‘‘๐‘ฅใ€— = (logโก๐‘ฅ )^2 โˆซ1โ–’ใ€–๐‘ฅ .ใ€— ๐‘‘๐‘ฅโˆ’โˆซ1โ–’((๐‘‘(logโก๐‘ฅ )^2)/๐‘‘๐‘ฅ โˆซ1โ–’ใ€–๐‘ฅ .๐‘‘๐‘ฅใ€—) ๐‘‘๐‘ฅ = (logโก๐‘ฅ )^2 . ๐‘ฅ^2/2โˆ’โˆซ1โ–’(2(logโก๐‘ฅ ) 1/๐‘ฅ โˆซ1โ–’ใ€–๐‘ฅ .๐‘‘๐‘ฅใ€—) ๐‘‘๐‘ฅ = ๐‘ฅ^2/2 (logโก๐‘ฅ )^2โˆ’2โˆซ1โ–’ใ€–logโก๐‘ฅ/๐‘ฅ . ๐‘ฅ^2/2ใ€— ๐‘‘๐‘ฅ = ๐‘ฅ^2/2 (logโก๐‘ฅ )^2โˆ’โˆซ1โ–’ใ€–๐‘ฅ logโก๐‘ฅ ใ€— ๐‘‘๐‘ฅ Now taking I1 = โˆซ1โ–’ใ€–๐‘ฅ logโก๐‘ฅ ใ€— ๐‘‘๐‘ฅ 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) = log x and g(x) = x โˆซ1โ–’ใ€–๐‘ฅ logโก๐‘ฅ ใ€— ๐‘‘๐‘ฅ=โˆซ1โ–’(logโก๐‘ฅ )๐‘ฅ ๐‘‘๐‘ฅ =logโก๐‘ฅ โˆซ1โ–’๐‘ฅ ๐‘‘๐‘ฅโˆ’โˆซ1โ–’(๐‘‘(logโก๐‘ฅ )/๐‘‘๐‘ฅ โˆซ1โ–’ใ€–๐‘ฅ.๐‘‘๐‘ฅใ€—)๐‘‘๐‘ฅ =logโก๐‘ฅ (๐‘ฅ^2/2)โˆ’โˆซ1โ–’ใ€–1/๐‘ฅ . ๐‘ฅ^2/2. ๐‘‘๐‘ฅใ€— =ใ€–๐‘ฅ^2/2 logใ€—โกใ€– ๐‘ฅใ€—โˆ’1/2 โˆซ1โ–’ใ€–๐‘ฅ. ๐‘‘๐‘ฅใ€— =ใ€–๐‘ฅ^2/2 logใ€—โก๐‘ฅโˆ’1/2 . ๐‘ฅ^2/2 +๐ถ =ใ€–๐‘ฅ^2/2 ๐‘™๐‘œ๐‘”ใ€—โกใ€– ๐‘ฅใ€—โˆ’ ๐‘ฅ^2/4 +๐ถ Putting the value of I1 in (1), we get โˆซ1โ–’ใ€–๐‘ฅ(logโก๐‘ฅ )^2.๐‘‘๐‘ฅใ€—=๐‘ฅ^2/2 (logโก๐‘ฅ )^2โˆ’โˆซ1โ–’ใ€– ๐‘ฅ .logโก๐‘ฅ ๐‘‘๐‘ฅใ€— =๐‘ฅ^2/2 (logโก๐‘ฅ )^2โˆ’((๐‘ฅ^2 (logโก๐‘ฅ ))/2 โˆ’ ๐‘ฅ^2/4 +๐ถ1) =๐’™^๐Ÿ/๐Ÿ (๐’๐’๐’ˆโก๐’™ )^๐Ÿโˆ’ (๐’™^๐Ÿ (๐’๐’๐’ˆโก๐’™ ))/๐Ÿ + ๐’™^๐Ÿ/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.