Example 27 - Chapter 7 Class 12 Integrals - Part 7

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Example 27 - Chapter 7 Class 12 Integrals - Part 8

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Example 27 - Chapter 7 Class 12 Integrals - Part 9 Example 27 - Chapter 7 Class 12 Integrals - Part 10

  1. Chapter 7 Class 12 Integrals (Term 2)
  2. Serial order wise

Transcript

Example 27 Evaluate the following integrals: (iii) โˆซ_1^2โ–’(๐‘ฅ ๐‘‘๐‘ฅ)/((๐‘ฅ + 1) (๐‘ฅ + 2) ) ๐‘‘๐‘ฅ ๐น(๐‘ฅ)=โˆซ1โ–’(๐‘ฅ ๐‘‘๐‘ฅ)/(๐‘ฅ + 1)(๐‘ฅ + 2) We can write the integrate as : ๐‘ฅ/(๐‘ฅ + 1)(๐‘ฅ + 2) =A/(๐‘ฅ + 1)+B/(๐‘ฅ + 2) ๐‘ฅ/(๐‘ฅ + 1)(๐‘ฅ + 2) =(A(๐‘ฅ + 2) + B(๐‘ฅ + 1))/(๐‘ฅ + 1)(๐‘ฅ + 2) Canceling denominators ๐‘ฅ=A(๐‘ฅ+2)+B(๐‘ฅ+1) Putting ๐’™=โˆ’๐Ÿ โˆ’2=A(2+2)+B(2+1) โˆ’2=A ร—0+B(โˆ’1) โˆ’2=โˆ’B B=2 Putting ๐’™=โˆ’๐Ÿ โˆ’1=A(โˆ’1+2)+B(โˆ’1+1) โˆ’1=A(1)+B(0) โˆ’1=A A=โˆ’1 Therefore ๐‘ฅ/(๐‘ฅ + 1)(๐‘ฅ + 2) =(โˆ’1)/(๐‘ฅ + 1)+2/(๐‘ฅ + 2) Integrating w.r.t.๐‘ฅ โˆซ1โ–’ใ€–๐‘ฅ/(๐‘ฅ+1)(๐‘ฅ+2) =โˆซ1โ–’ใ€–(โˆ’1)/((๐‘ฅ+1) ) ๐‘‘๐‘ฅใ€—+โˆซ1โ–’ใ€–2/(๐‘ฅ+2) ๐‘‘๐‘ฅใ€—ใ€— =โˆ’๐‘™๐‘œ๐‘”|๐‘ฅ+1|+2๐‘™๐‘œ๐‘”|๐‘ฅ+2| =โˆ’๐‘™๐‘œ๐‘”|๐‘ฅ+1|+๐‘™๐‘œ๐‘”|๐‘ฅ+2|^2 =๐‘™๐‘œ๐‘”|๐‘ฅ+2|^2โˆ’๐‘™๐‘œ๐‘”|๐‘ฅ+1| =๐‘™๐‘œ๐‘”|(๐‘ฅ + 2)^2/(๐‘ฅ + 1)| Hence ๐น(๐‘ฅ)=๐‘™๐‘œ๐‘”|(๐‘ฅ + 2)^2/(๐‘ฅ + 1)| Now, โˆซ_1^2โ–’ใ€–๐‘ฅ/(๐‘ฅ + 1)(๐‘ฅ + 2) ๐‘‘๐‘ฅใ€—=๐น(2)โˆ’๐น(1) โˆซ_1^2โ–’ใ€–๐‘ฅ/(๐‘ฅ + 1)(๐‘ฅ + 2) ๐‘‘๐‘ฅใ€—=๐‘™๐‘œ๐‘”|(2 + 2)^2/(2 + 1)|โˆ’๐‘™๐‘œ๐‘”|(1 + 2)^2/(1 + 1)| (๐‘ logโกใ€–๐‘Ž=logโกใ€–๐‘Ž^๐‘ ใ€— ใ€— ) (logโกAโˆ’logโกB=๐‘™๐‘œ๐‘” A/๐ต) =๐‘™๐‘œ๐‘”|(4)^2/3|โˆ’๐‘™๐‘œ๐‘”|(3)^2/2| =๐‘™๐‘œ๐‘”|(4^2/3)/(3^2/2)| =๐‘™๐‘œ๐‘”|4^2/3 ร— 2/3^2 | =๐‘™๐‘œ๐‘”|16/3 ร— 2/9| =๐‘™๐‘œ๐‘”|32/27 | =๐ฅ๐จ๐ โก(๐Ÿ‘๐Ÿ/๐Ÿ๐Ÿ•) (logโกAโˆ’logโกB=๐‘™๐‘œ๐‘” A/๐ต) (logโกAโˆ’logโกB=๐‘™๐‘œ๐‘” A/๐ต)

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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.