Find ∫ x 2 + 1 / (x 2 + 2) (x 2 + 3) dx

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

Transcript

Question 33 Find โˆซ1โ–’ใ€–(๐‘ฅ^2 + 1)/((๐‘ฅ^2 + 2) (๐‘ฅ^2 + 3)) ๐‘‘๐‘ฅใ€— Putting ๐’™^๐Ÿ=๐’š (๐‘ฅ^2 + 1 )/((๐‘ฅ^2 + 2) (๐‘ฅ^2 + 3) )=(๐‘ฆ + 1)/((๐‘ฆ + 2) (๐‘ฆ + 3) ) We can write this in form (๐‘ฆ + 1)/((๐‘ฆ + 2) (๐‘ฆ + 3) )=๐ด/((๐‘ฆ + 2) ) + ๐ต/((๐‘ฆ + 3) ) (๐‘ฆ + 1)/((๐‘ฆ + 2) (๐‘ฆ + 3) )=(๐ด(๐‘ฆ +3) + ๐ต (๐‘ฆ + 2))/((๐‘ฆ + 2) (๐‘ฆ + 3) ) By cancelling denominator ๐‘ฆ+1=๐ด(๐‘ฆ +3) + ๐ต (๐‘ฆ + 2) Putting y = โˆ’3 โˆ’3+1=๐ด(โˆ’3+3)+๐ต(โˆ’3+2) โˆ’2=๐ด ร— 0+๐ต ร— โˆ’1 โˆ’2=โˆ’๐ต ๐‘ฉ=๐Ÿ Putting y = โˆ’2 โˆ’2+1=๐ด(โˆ’2+3)+๐ต(โˆ’2+2) โˆ’1=๐ด ร— 1+๐ต ร— 0 โˆ’1=๐ด ๐‘จ=โˆ’๐Ÿ Hence we can write (๐‘ฆ + 1)/((๐‘ฆ + 2) (๐‘ฆ + 3) )=(โˆ’1)/((๐‘ฆ + 2) ) + 2/((๐‘ฆ + 3) ) Substituting back ๐‘ฆ=๐‘ฅ^2 (๐‘ฅ^2 + 1 )/((๐‘ฅ^2 + 2) (๐‘ฅ^2 + 3) ) =(โˆ’1)/((๐‘ฅ^2 + 2) )+2/((๐‘ฅ^2 + 3) ) Therefore, โˆซ1โ–’(๐‘ฅ^2 + 1 )/((๐‘ฅ^2 + 2) (๐‘ฅ^2 + 3) ) ๐‘‘๐‘ฅ=โˆซ1โ–’(โˆ’1)/((๐‘ฅ^2 + 2) ) ๐‘‘๐‘ฅ+โˆซ1โ–’2/((๐‘ฅ^2 + 3) ) ๐‘‘๐‘ฅ =โˆ’โˆซ1โ–’1/((๐‘ฅ^2 +(โˆš2)^2 ) ) ๐‘‘๐‘ฅ+2โˆซ1โ–’1/((๐‘ฅ^2 +(โˆš3)^2 ) ) ๐‘‘๐‘ฅ By using formula โˆซ1โ–’1/(๐‘ฅ^2 + ๐‘Ž^2 ) ๐‘‘๐‘ฅ=1/๐‘Ž ใ€–๐‘ก๐‘Ž๐‘›ใ€—^(โˆ’1)โก(๐‘ฅ/๐‘Ž)+๐ถ =(โˆ’๐Ÿ)/โˆš๐Ÿ ใ€–๐’•๐’‚๐’ใ€—^(โˆ’๐Ÿ)โกใ€–๐’™/โˆš๐Ÿใ€—+๐Ÿ/โˆš๐Ÿ‘ ใ€–๐’•๐’‚๐’ใ€—^(โˆ’๐Ÿ)โกใ€–๐’™/โˆš๐Ÿ‘ใ€— +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 9 years. He provides courses for Maths and Science at Teachoo.