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Ex 7.6, 14 Class 12 Maths - NCERT Solutions - Integrate x (log x)^2

Ex 7.6, 14 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.6, 14 - Chapter 7 Class 12 Integrals - Part 3

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Ex 7.6, 14 ใ€–๐‘ฅ(logโก๐‘ฅ)ใ€—^2 โˆซ1โ–’ใ€–๐‘ฅ(logโก๐‘ฅ )^2.๐‘‘๐‘ฅ " " ใ€— 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 Now 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) = x and g(x) = (log x)2 = 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 Solving I1 I1 = โˆซ1โ–’ใ€–๐‘ฅ logโก๐‘ฅ ใ€— ๐‘‘๐‘ฅ 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 Now 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) = x and g(x) = log x =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 value of I1 in (1), โˆซ1โ–’ใ€–๐‘ฅ(logโก๐‘ฅ )^2.๐‘‘๐‘ฅใ€—=๐‘ฅ^2/2 (logโก๐‘ฅ )^2โˆ’โˆซ1โ–’ใ€– ๐’™ .๐’๐’๐’ˆโก๐’™ ๐’…๐’™ใ€— =๐‘ฅ^2/2 (logโก๐‘ฅ )^2โˆ’((๐‘ฅ^2 (logโก๐‘ฅ ))/2 โˆ’ ๐‘ฅ^2/4 +๐ถ1) =๐‘ฅ^2/2 (logโก๐‘ฅ )^2โˆ’ (๐‘ฅ^2 (logโก๐‘ฅ ))/2 + ๐‘ฅ^2/4 โˆ’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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.