Example 22 - Find (i) ex ( tan-1 x + 1 / 1 + x2) dx - Examples

Example 22 - Chapter 7 Class 12 Integrals - Part 2
Example 22 - Chapter 7 Class 12 Integrals - Part 3 Example 22 - Chapter 7 Class 12 Integrals - Part 4 Example 22 - Chapter 7 Class 12 Integrals - Part 5

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Transcript

Example 22 Find (i) โˆซ1โ–’๐‘’^๐‘ฅ (tan^(โˆ’1)โก๐‘ฅ+ 1/(1 + ๐‘ฅ^2 )) ๐‘‘๐‘ฅ โˆซ1โ–’ใ€–๐‘’^๐‘ฅ (tan^(โˆ’1)โก๐‘ฅ+1/(1 + ๐‘ฅ^2 ))๐‘‘๐‘ฅใ€— It is of the form โˆซ1โ–’ใ€–๐‘’^๐‘ฅ [๐‘“(๐‘ฅ)+๐‘“^โ€ฒ (๐‘ฅ)] ใ€— ๐‘‘๐‘ฅ=๐‘’^๐‘ฅ ๐‘“(๐‘ฅ)+๐ถ Where ๐‘“(๐‘ฅ)=tan^(โˆ’1)โก๐‘ฅ ๐‘“^โ€ฒ (๐‘ฅ)= 1/(1 + ๐‘ฅ^2 ) So, our equation becomes โˆซ1โ–’ใ€–๐‘’^๐‘ฅ (tan^(โˆ’1)โก๐‘ฅ+1/(1 + ๐‘ฅ^2 ))๐‘‘๐‘ฅใ€—=๐’†^๐’™ ใ€–๐ญ๐š๐งใ€—^(โˆ’๐Ÿ)โกใ€–๐’™+๐‘ชใ€— Example 22 Find (ii) โˆซ1โ–’((๐‘ฅ^2 + 1) ๐‘’^๐‘ฅ)/(๐‘ฅ + 1)^2 ๐‘‘๐‘ฅ โˆซ1โ–’ใ€–(๐‘ฅ^2 + 1)/(๐‘ฅ + 1)^2 .๐‘’^๐‘ฅ ๐‘‘๐‘ฅใ€— Adding and subtracting 1 in numerator =โˆซ1โ–’ใ€–(๐‘ฅ^2+ 1 + 1 โˆ’ 1)/(๐‘ฅ + 1)^2 .๐‘’^๐‘ฅ .๐‘‘๐‘ฅใ€— =โˆซ1โ–’ใ€–(๐‘ฅ^2 โˆ’ 1 + 1 + 1)/(๐‘ฅ + 1)^2 .๐‘’^๐‘ฅ .๐‘‘๐‘ฅใ€— =โˆซ1โ–’ใ€–[(๐‘ฅ^2 โˆ’ 1)/(๐‘ฅ + 1)^2 +2/(๐‘ฅ + 1)^2 ] ๐‘’^๐‘ฅ ๐‘‘๐‘ฅใ€— =โˆซ1โ–’ใ€–๐‘’^๐‘ฅ [(๐‘ฅ^2 โˆ’ (1)^2)/(๐‘ฅ + 1)^2 +2/(๐‘ฅ + 1)^2 ]๐‘‘๐‘ฅใ€— =โˆซ1โ–’ใ€–๐‘’^๐‘ฅ [(๐‘ฅ โˆ’ 1)(๐‘ฅ + 1)/(๐‘ฅ + 1)^2 +2/(๐‘ฅ + 1)^2 ]๐‘‘๐‘ฅใ€— =โˆซ1โ–’ใ€–๐‘’^๐‘ฅ [(๐‘ฅ โˆ’ 1)/(๐‘ฅ + 1)+2/(๐‘ฅ + 1)^2 ]๐‘‘๐‘ฅใ€— It is of form โˆซ1โ–’ใ€–๐‘’^๐‘ฅ [๐‘“(๐‘ฅ)+๐‘“^โ€ฒ (๐‘ฅ)] ใ€— ๐‘‘๐‘ฅ=๐‘’^๐‘ฅ ๐‘“(๐‘ฅ)+๐ถ Where ๐‘“(๐‘ฅ)=(๐‘ฅ โˆ’ 1)/(๐‘ฅ + 1) ๐‘“^โ€ฒ (๐‘ฅ)=๐‘‘/๐‘‘๐‘ฅ [(๐‘ฅ โˆ’ 1)/(๐‘ฅ + 1)] ๐‘“^โ€ฒ (๐‘ฅ)=(1.(๐‘ฅ + 1) โˆ’1 (๐‘ฅ โˆ’ 1))/(๐‘ฅ + 1)^2 =(๐‘ฅ + 1 โˆ’ ๐‘ฅ + 1)/(๐‘ฅ + 1)^2 =2/(๐‘ฅ + 1)^2 Thus, our equation becomes โˆซ1โ–’ใ€–(๐‘ฅ^2 + 1)/(๐‘ฅ + 1)^2 .๐‘’^๐‘ฅ=โˆซ1โ–’ใ€–๐‘’^๐‘ฅ [(๐‘ฅ โˆ’ 1)/(๐‘ฅ + 1)+2/(๐‘ฅ + 1)^2 ]๐‘‘๐‘ฅใ€—ใ€— =๐‘’^๐‘ฅ [(๐‘ฅ โˆ’ 1)/(๐‘ฅ + 1)]+๐ถ =(๐’™ โˆ’ ๐Ÿ)/(๐’™ + ๐Ÿ).๐’†^๐’™+๐‘ช

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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.