Misc 32 - Chapter 7 Class 12 Integrals

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Misc 32 Prove that โˆซ_1^3โ–’ใ€–๐‘‘๐‘ฅ/(๐‘ฅ^2 (๐‘ฅ + 1) )= 2/3ใ€—+logโกใ€–2/3ใ€— Solving L.H.S : โˆซ_1^3โ–’๐‘‘๐‘ฅ/(๐‘ฅ^2 (๐‘ฅ + 1) ) By partial fraction, 1/(๐‘ฅ^2 (๐‘ฅ + 1)) = A/๐‘ฅ+B/๐‘ฅ^2 +C/(๐‘ฅ + 1) 1/(๐‘ฅ^2 (๐‘ฅ + 1)) = ( A ๐‘ฅ (๐‘ฅ + 1) + B (๐‘ฅ + 1) + C๐‘ฅ^2)/(๐‘ฅ^2 (๐‘ฅ + 1)) โˆด 1 = Ax (x + 1) + B (x + 1) + C๐‘ฅ^2 Finding A,B,C โˆด 1/(๐‘ฅ^2 (๐‘ฅ + 1))= (โˆ’1)/๐‘ฅ+1/๐‘ฅ^2 +1/(๐‘ฅ + 1) โˆซ1โ–’1/(๐‘ฅ^2 (๐‘ฅ + 1)) ๐‘‘๐‘ฅ=โˆซ1โ–’(โˆ’1)/๐‘ฅ+1/๐‘ฅ^2 +1/(๐‘ฅ + 1) ๐‘‘๐‘ฅ = [โˆ’log|๐‘ฅ|โˆ’1/๐‘ฅ+log|๐‘ฅ+1|]_1^3 = [log|(๐‘ฅ + 1)/๐‘ฅ|โˆ’1/๐‘ฅ]_1^3 Putting the limits = [log(4/3)โˆ’1/3]โˆ’[log(2)โˆ’1] = "log" 4/3โˆ’"log" 2 โˆ’ 1/3+1 = "log" (4/3ร—1/2)โˆ’1/3+1 = ๐‘™๐‘œ๐‘”(2/3)+2/3 = R.H.S Hence, proved.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo