Slide32.JPG

Slide33.JPG
Slide34.JPG
Slide35.JPG
Slide36.JPG


Transcript

Misc 23 Integrate the function (โˆš(๐‘ฅ^2 + 1) [logโกใ€–(๐‘ฅ^2+ 1) โˆ’ 2 logโก๐‘ฅ ใ€— ] )/๐‘ฅ^4 โˆซ1โ–’(โˆš(๐‘ฅ^2 + 1) [logโกใ€–(๐‘ฅ^2+ 1) โˆ’ 2 logโก๐‘ฅ ใ€— ] )/๐‘ฅ^4 ๐‘‘๐‘ฅ Taking ๐‘ฅ^2common from โˆš(๐‘ฅ^2+1) = โˆซ1โ–’(ใ€–ใ€–(๐‘ฅใ€—^2) ใ€—^(1/2) (1 + 1/๐‘ฅ^2 )^(1/2) (logโกใ€–(๐‘ฅ^2+1)ใ€— โˆ’ logโกใ€–๐‘ฅ^2 ใ€— ))/๐‘ฅ^4 ๐‘‘๐‘ฅ = โˆซ1โ–’(๐‘ฅ (1+ 1/๐‘ฅ^2 )^(1/2) (logโกใ€– ((๐‘ฅ^(2 )+ 1))/๐‘ฅ^2 ใ€— ))/๐‘ฅ^4 ๐‘‘๐‘ฅ = โˆซ1โ–’( (1+ 1/๐‘ฅ^2 )^(1/2) (logโก(1 + 1/๐‘ฅ^2 ) ))/๐‘ฅ^3 Let t = 1 + 1/๐‘ฅ^2 ๐‘‘๐‘ก/๐‘‘๐‘ฅ=(โˆ’2)/๐‘ฅ^3 (โˆ’1)/2 ๐‘‘๐‘ก=๐‘‘๐‘ฅ/๐‘ฅ^3 Substituting, = โˆ’1/2 โˆซ1โ–’๐‘ก^(1/2) ใ€– log ๐‘กใ€—โกใ€– ๐‘‘๐‘กใ€— = โˆซ1โ–’( (1+ 1/๐‘ฅ^2 )^(1/2) (logโก(1 + 1/๐‘ฅ^2 ) ))/๐‘ฅ^3 Let t = 1 + 1/๐‘ฅ^2 ๐‘‘๐‘ก/๐‘‘๐‘ฅ=(โˆ’2)/๐‘ฅ^3 (โˆ’1)/2 ๐‘‘๐‘ก=๐‘‘๐‘ฅ/๐‘ฅ^3 Substituting value of t and dt = (โˆ’1)/2 โˆซ1โ–’๐‘ก^(1/2) ใ€– log ๐‘กใ€—โกใ€– ๐‘‘๐‘กใ€— Hence, (โˆ’1)/2 โˆซ1โ–’ใ€–๐‘ก^(1/2) logโกใ€–๐‘ก ๐‘‘๐‘ก=(โˆ’1)/2 (logโกใ€–๐‘ก โˆซ1โ–’ใ€–๐‘ก^(1/2) ๐‘‘๐‘กใ€—โˆ’โˆซ1โ–’((๐‘‘(logโกใ€–๐‘ก)ใ€—)/๐‘‘๐‘ก โˆซ1โ–’๐‘ก^(1/2) ๐‘‘๐‘ก) ๐‘‘๐‘กใ€— )ใ€— ใ€— = (โˆ’1)/2 (logโกใ€–๐‘ก (๐‘ก^(3/2)/(3/2))โˆ’โˆซ1โ–’ใ€–1/๐‘กร—(๐‘ก^(3/2)/(3/2)) ใ€—ใ€— ๐‘‘๐‘ก) = (โˆ’1)/2 (2/3 ๐‘ก^(3/2) logโกใ€–๐‘กโˆ’2/3ใ€— โˆซ1โ–’ใ€–๐‘ก^(1/2) ๐‘‘๐‘กใ€—) = (โˆ’1)/2 (2/3 ๐‘ก^(3/2) logโกใ€–๐‘กโˆ’2/3ใ€— ( ใ€–2๐‘กใ€—^(3/2))/3) = (โˆ’1)/3 ๐‘ก^(3/2) logโก๐‘ก + 2/9 ๐‘ก^(3/2) Putting value of t = 1 + 1/๐‘ฅ^2 = (โˆ’1)/3 (1+1/๐‘ฅ^2 )^(3/2) logโกใ€–(1+1/๐‘ฅ^2 )+2/9 " " (1+1/๐‘ฅ^2 )^(3/2)+ใ€— C = (โˆ’๐Ÿ)/๐Ÿ‘ (๐Ÿ+๐Ÿ/๐’™^(๐Ÿ ) )^(๐Ÿ‘/๐Ÿ) (๐ฅ๐จ๐ โก(๐Ÿ+๐Ÿ/๐’™^๐Ÿ )โˆ’๐Ÿ/๐Ÿ‘)+ C

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

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, Social Science, Physics, Chemistry, Computer Science at Teachoo.