# Misc 29 - Chapter 7 Class 12 Integrals

Last updated at Dec. 23, 2019 by Teachoo

Last updated at Dec. 23, 2019 by Teachoo

Transcript

Misc 29 Evaluate the definite integral โซ_0^1โใ๐๐ฅ/(โ(1 + ๐ฅ) โ โ๐ฅ) ใ โซ_0^1โใ๐๐ฅ/(โ(1 + ๐ฅ) โ โ๐ฅ) ใ Rationalizing i.e., multiplying and dividing by (โ(1 + ๐ฅ)+โ๐ฅ) = โซ_0^1โใ๐๐ฅ/(โ(1 + ๐ฅ) โ โ๐ฅ) ใร(โ(1 + ๐ฅ) + โ๐ฅ)/(โ(1 + ๐ฅ) + โ๐ฅ) . ๐๐ฅ = โซ_0^1โ(โ(1 + ๐ฅ) + โ๐ฅ)/((โ(1 + ๐ฅ) )^2 โ (โ๐ฅ )^2 ) . ๐๐ฅ = โซ_0^1โ(โ(1 + ๐ฅ) + โ๐ฅ)/(1 + ๐ฅ โ ๐ฅ) . ๐๐ฅ = โซ_0^1โ(โ(1 + ๐ฅ) + โ๐ฅ)/1 . ๐๐ฅ = โซ_0^1โโ(1+๐ฅ) . ๐๐ฅ+โซ_0^1โโ๐ฅ . ๐๐ฅ" " = โซ_0^1โ(1+๐ฅ)^(1/2) ๐๐ฅ+โซ_0^1โ(๐ฅ)^(1/2) ๐๐ฅ" " = [(1 + ๐ฅ)^(1/2 + 1)/(1/2 + 1)]_0^1 + [ใ๐ฅ ใ^(1/2 + 1)/(1/2 + 1)]_0^1 = [(1 + ๐ฅ)^(3/2)/(3/2)]_0^1 + [ใ๐ฅ ใ^(3/2)/(3/2)]_0^1 = ใ2/3 [(1+๐ฅ)^(3/2) ]ใ_0^1 + 2/3 [ใ๐ฅ ใ^(3/2) ]_0^1 = 2/3 [(1+1)^(3/2)โ(1+0)^(3/2) ] + 2/3 [(1)^(3/2)โ(0)^(3/2) ] = 2/3 [(2)^(3/2)โ(1)^(3/2) ] + 2/3 [1โ0] = 2/3 . (2)^(3/2)โ2/3 [1]+2/3 [1] = 2/3 (2)^(3/2) = 2/3 [(2)^(1/2) ]^3 = 2/3 (โ2 )^3 = 2/3 . 2 โ2 = (๐ โ๐)/๐

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.