If 5+7 -2 ^2 d =A _e|f( )|+c (where c is a constant of integration), then f( ) A can be

Step-by-step solution

aligned & Let I= 5+7 -2 ^2 d & = 5+7 -2 (1- ^2 ) d & = 2 ^2 +7 +3 d & = ( +3)(2 +1) d & Put =t & d =d t aligned aligned I & = d t (t+3)(2 t+1) & = [ 2 5(2 t+1) - 1 5(t+3) ] d t & = 2 5 |2 t+1| 2 - 1 5 |t+3|+c & = 1 5 |2 +1|- 1 5 | +3|+c & = 1 5 | 2 +1 +3 |+c aligned aligned & A = 1 5 , f ( )= 2 +1 +3 & f ( ) A = 5(2 +1) +3 aligned