Find f log x (1 + log x) 2 dx

This question is similar to Ex 7.2, 35 - Chapter 7 Class 12 - Integrals

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  1. Class 12
  2. Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Transcript

Question 1 โ€“ Choice 1 Find โˆซ1โ–’ใ€–logโก๐‘ฅ/(1 + logโก๐‘ฅ )^2 ๐‘‘๐‘ฅใ€— Let ๐ˆ=โˆซ1โ–’ใ€–logโก๐‘ฅ/(1 + logโก๐‘ฅ )^2 ๐‘‘๐‘ฅใ€— =โˆซ1โ–’ใ€–(๐’๐’๐’ˆโก๐’™ + ๐Ÿ โˆ’ ๐Ÿ)/(1 + logโก๐‘ฅ )^2 ๐‘‘๐‘ฅใ€— =โˆซ1โ–’ใ€–((1 + logโก๐‘ฅ) โˆ’ 1)/(1 + logโก๐‘ฅ )^2 ๐‘‘๐‘ฅใ€— =โˆซ1โ–’ใ€–((1 + logโก๐‘ฅ) )/(1 + logโก๐‘ฅ )^2 ๐‘‘๐‘ฅใ€—โˆ’โˆซ1โ–’ใ€–1/(1 + logโก๐‘ฅ )^2 ๐‘‘๐‘ฅใ€— =โˆซ1โ–’ใ€–(๐Ÿ )/((๐Ÿ + ๐’๐’๐’ˆโก๐’™ ) ) ๐’…๐’™ใ€—โˆ’โˆซ1โ–’ใ€–๐Ÿ/(๐Ÿ + ๐’๐’๐’ˆโก๐’™ )^๐Ÿ ๐’…๐’™ใ€— Solving โˆซ1โ–’ใ€–(๐Ÿ )/((๐Ÿ + ๐ฅ๐จ๐ โก๐ฑ ) ) ๐๐ฑใ€— Using Integration by parts โˆซ1โ–’ใ€–(๐Ÿ )/((๐Ÿ + ๐’๐’๐’ˆโก๐’™ ) ) ๐’…๐’™ใ€— = โˆซ1โ–’ใ€–(1 )/((1 + ๐‘™๐‘œ๐‘”โก๐‘ฅ ) ) ร— 1 ๐‘‘๐‘ฅใ€— = 1/((1 + logโก๐‘ฅ)) โˆซ1โ–’ใ€–1 ๐‘‘๐‘ฅใ€—โˆ’โˆซ1โ–’(๐’…(๐Ÿ/(๐Ÿ + ๐’๐’๐’ˆ ๐’™))/๐’…๐’™ โˆซ1โ–’ใ€–๐Ÿ ๐’…๐’™ใ€—) ๐’…๐’™ = 1/((1 + logโก๐‘ฅ))ร— ๐‘ฅโˆ’โˆซ1โ–’((โˆ’๐Ÿ)/(๐Ÿ +๐’๐’๐’ˆ ๐’™)^๐Ÿ ร—๐Ÿ/๐’™ ร— ๐’™) ๐’…๐’• = ๐‘ฅ/((1 + logโก๐‘ฅ))+โˆซ1โ–’๐Ÿ/(๐Ÿ + ๐’๐’๐’ˆ ๐’™)^๐Ÿ ๐’…๐’™ Thus I=โˆซ1โ–’ใ€–(1 )/((1 + ๐‘™๐‘œ๐‘”โก๐‘ฅ ) ) ๐‘‘๐‘ฅใ€—โˆ’โˆซ1โ–’ใ€–1/(1 + ๐‘™๐‘œ๐‘”โก๐‘ฅ )^2 ๐‘‘๐‘ฅใ€— = ๐‘ฅ/((1 + logโก๐‘ฅ))+โˆซ1โ–’1/(1 + ๐‘™๐‘œ๐‘” ๐‘ฅ)^2 ๐‘‘๐‘ฅโˆ’โˆซ1โ–’ใ€–1/(1 + ๐‘™๐‘œ๐‘”โก๐‘ฅ )^2 ๐‘‘๐‘ฅใ€— = ๐’™/((๐Ÿ + ๐’๐’๐’ˆโก๐’™))+๐‘ช

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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.