Check sibling questions

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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Ex 7.8, 8 โˆซ_(๐œ‹/6)^(๐œ‹/4)โ–’ใ€–๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅใ€—โก๐‘‘๐‘ฅ Let F(๐‘ฅ)=โˆซ1โ–’ใ€–๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ . ๐‘‘๐‘ฅใ€— Multiplying and Dividing by ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ F(๐‘ฅ)=โˆซ1โ–’(๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ (๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ))/(๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ) ๐‘‘๐‘ฅ Let c๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ=๐‘ก Differentiating w.r.t. ๐‘ฅ ๐‘‘/๐‘‘๐‘ฅ (๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ)=๐‘‘๐‘ก/๐‘‘๐‘ฅ โˆ’๐‘๐‘œ๐‘ ๐‘’๐‘^2 ๐‘ฅโˆ’๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ ๐‘๐‘œ๐‘ก ๐‘ฅ=๐‘‘๐‘ก/๐‘‘๐‘ฅ โˆ’๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ(๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ)=๐‘‘๐‘ก/๐‘‘๐‘ฅ ๐‘‘๐‘ฅ=๐‘‘๐‘ก/(โˆ’๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ (๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ) ) Therefore, โˆซ1โ–’(๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ(๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ))/(๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ) ๐‘‘๐‘ฅ =โˆซ1โ–’(๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ(๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ))/๐‘ก. ๐‘‘๐‘ก/(โˆ’๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ(๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ) ) =โˆ’โˆซ1โ–’๐‘‘๐‘ก/๐‘ก =โˆ’logโกใ€– |๐‘ก|ใ€— =โˆ’logโกใ€– |๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ|ใ€— Hence, F(๐‘ฅ)=โˆ’logโก|๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+cotโก๐‘ฅ | Now, โˆซ_(๐œ‹/6)^(๐œ‹/4)โ–’ใ€–๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ=๐น(๐œ‹/4)โˆ’๐น(๐œ‹/6) ใ€— =โˆ’๐‘™๐‘œ๐‘”|๐‘๐‘œ๐‘ ๐‘’๐‘(๐œ‹/4)+๐‘๐‘œ๐‘ก(๐œ‹/4)|โˆ’ (โˆ’logโก|๐‘๐‘œ๐‘ ๐‘’๐‘(๐œ‹/6)+ ๐‘๐‘œ๐‘ก(๐œ‹/6)| ) =โˆ’๐‘™๐‘œ๐‘”|โˆš2+1|+๐‘™๐‘œ๐‘”|2+โˆš3| =โˆ’๐‘™๐‘œ๐‘”|2+โˆš3|โˆ’ ๐‘™๐‘œ๐‘”|โˆš2+1| =๐‘™๐‘œ๐‘”|(2 + โˆš3)/(โˆš2 + 1)| =๐‘™๐‘œ๐‘”|(2 + โˆš3)/(โˆš2 + 1)ร—(2 โˆ’ โˆš3)/(2 โˆ’ โˆš3)| =๐‘™๐‘œ๐‘”[((2)^2 โˆ’ (โˆš3)^2)/((โˆš2 + 1) ร— (2 โˆ’ โˆš3) )] =๐‘™๐‘œ๐‘”[(4 โˆ’ 3)/(โˆš2 + 1)(2 โˆ’ โˆš3) ] =๐‘™๐‘œ๐‘”[(1 ร— (โˆš2 โˆ’ 1))/((2 โˆ’ โˆš3)(โˆš2 + 1) ร—(โˆš2 โˆ’ 1) )] =๐‘™๐‘œ๐‘”[(โˆš2 โˆ’ 1)/(2 โˆ’ โˆš3)[(โˆš2)^2โˆ’ 1^2 ] ] =๐’๐’๐’ˆ[(โˆš๐Ÿ โˆ’ ๐Ÿ)/(๐Ÿ โˆ’ โˆš๐Ÿ‘)]

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Davneet Singh

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.