Misc 32 - Chapter 7 Class 12 Integrals
Last updated at April 16, 2024 by Teachoo
Definite Integration - By Partial Fraction
Definite Integration - By Partial Fraction
Last updated at April 16, 2024 by Teachoo
Misc 32 Prove that โซ_1^3โใ๐๐ฅ/(๐ฅ^2 (๐ฅ + 1) )= 2/3ใ+logโกใ2/3ใ Solving L.H.S : โซ_1^3โ๐๐ฅ/(๐ฅ^2 (๐ฅ + 1) ) By partial fraction, 1/(๐ฅ^2 (๐ฅ + 1)) = A/๐ฅ+B/๐ฅ^2 +C/(๐ฅ + 1) 1/(๐ฅ^2 (๐ฅ + 1)) = ( A ๐ฅ (๐ฅ + 1) + B (๐ฅ + 1) + C๐ฅ^2)/(๐ฅ^2 (๐ฅ + 1)) โด 1 = Ax (x + 1) + B (x + 1) + C๐ฅ^2 Finding A,B,C โด 1/(๐ฅ^2 (๐ฅ + 1))= (โ1)/๐ฅ+1/๐ฅ^2 +1/(๐ฅ + 1) โซ1โ1/(๐ฅ^2 (๐ฅ + 1)) ๐๐ฅ=โซ1โ(โ1)/๐ฅ+1/๐ฅ^2 +1/(๐ฅ + 1) ๐๐ฅ = [โlog|๐ฅ|โ1/๐ฅ+log|๐ฅ+1|]_1^3 = [log|(๐ฅ + 1)/๐ฅ|โ1/๐ฅ]_1^3 Putting the limits = [log(4/3)โ1/3]โ[log(2)โ1] = "log" 4/3โ"log" 2 โ 1/3+1 = "log" (4/3ร1/2)โ1/3+1 = ๐๐๐(2/3)+2/3 = R.H.S Hence, proved.