Chapter 7 Class 12 Integrals
Concept wise

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Transcript

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 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.