# Ex 7.9, 8 - Chapter 7 Class 12 Integrals

Last updated at April 16, 2024 by Teachoo

Last updated at April 16, 2024 by Teachoo

Ex 7.9, 8 Evaluate the integrals using substitution โซ_1^(2 )โใ (1/๐ฅ โ1/(2๐ฅ^2 )) ใ ๐^2๐ฅ ๐๐ฅ Let ๐ก=2๐ฅ ๐๐ก/๐๐ฅ=2 ๐๐ก/2=๐๐ฅ Thus, when x varies from 1 to 2, t varies from 2 to 4 Substituting, โซ_1^(2 )โใ (1/๐ฅ โ1/(2๐ฅ^2 )) ใ ๐^2๐ฅ ๐๐ฅ = โซ_2^4โใ๐^๐ก (1/(๐ก/2)โ1/(2ใ (๐ก/2)ใ^2 )) ใ ๐๐ก/2 =โซ_2^4โใ๐^๐ก (2/๐กโ4/(2๐ก^2 )) ใ ๐๐ก/2 =โซ_2^4โใ๐^๐ก (1/๐กโ2/๐ก^2 ) ใ ๐๐ก It is of the form โซ1โใ๐^๐ฅ [๐(๐ฅ)+๐^โฒ (๐ฅ)] ใ ๐๐ฅ=๐^๐ฅ ๐(๐ฅ)+๐ถ Where ๐(๐ฅ)=1/๐ก ๐^โฒ (๐ฅ)= (โ1)/๐ก^2 Hence, our equation becomes โซ_2^4โใ๐^๐ก (1/๐กโ2/๐ก^2 ) ใ ๐๐ก = [๐^๐กร1/๐ก]_2^4 = (๐^4/4โ๐^2/2) = (๐^๐ (๐^๐ โ ๐))/๐