Example 22 - Find (i) ex ( tan-1 x + 1 / 1 + x2) dx - Integration by parts - e^x integration

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Example 22 Find (i) 1 ^ (tan^( 1) + 1/(1 + ^2 )) 1 ^ [tan^( 1) +1/(1+ ^2 ) ] Putting ( )=tan^( 1) ^ ( )= 1/(1 + ^2 ) Replacing ( ) by (tan^( 1) ) and ^ ( ) by 1/(1 + ^2 ), we get 1 ^ [tan^( 1) +1/(1+ ^2 ) ] = ^ [tan^( 1) ]+ = ^ tan^( 1) + Example 22 Find (ii) 1 (( ^2 + 1) ^ )/( + 1)^2 Integrating 1 ( ^2 + 1)/( +1)^2 . ^ = 1 ( ^2+ 1 + 1 1)/( +1)^2 . ^ . = 1 ( ^2 1 + 1 + 1)/( +1)^2 . ^ . = 1 [( ^2 1)/( +1)^2 +2/( +1)^2 ] ^ = 1 [( ^2 (1)^2)/( +1)^2 +2/( +1)^2 ] = 1 [( 1)( +1)/( +1)^2 +2/( +1)^2 ] = 1 [( 1)( +1)/( +1)^2 +2/( +1)^2 ] = 1 [( 1)/( + 1)+2/( +1)^2 ] Putting ( )=( 1)/( + 1) ^ ( )= / [( 1)/( + 1)] Here, ( )= 1 ^ ( )=(1.( + 1) 1 ( 1))/( + 1)^2 ^ ( )=( + 1 + 1)/( + 1)^2 ^ ( )=2/( + 1)^2 Replacing ( ) by ( 1)/( + 1) and ^ ( ) by 2/( + 1)^2 , we get 1 ( ^2 + 1)/( + 1)^2 . = 1 [( 1)/( + 1)+2/( + 1)^2 ] = ^ [( 1)/( + 1)]+ =( 1)/( + 1). ^ +

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