If d x ^3 x 2 2 x =( x)^ A + C ( x)^ B + K , where K is a constant of integration, then the value of 5( ~A + B + C ) is equal to [Note: The question has been modified to get the correct answer.]

Step-by-step solution

aligned & Let I = d x ^3 x 2 2 x & Put 2 x= 2 x 1+ ^2 x & I = ^2 x ^2 x 2 x x ~d x aligned Let x= t ^2 ~d x= dt aligned I & = 1 2 1+t^2 t d t= 1 2 ( 1 t +t^ 3 2 ) d t & = 1 2 [ t^ 1 2 1 2 + t^ 5 2 5 2 ]+k=( )^ 1 2 + 1 5 ( )^ 5 2 +k aligned Comparing with ( x)^ A + C ( x)^ B + k , we get aligned & A= 1 2 , B= 5 2 , C= 1 5 & 5(A+B+C)=5 ( 1 2 + 5 2 + 1 5 )=16 aligned [Note: In the question, d x ^3 x x 2 x is changed to d x ^3 x 2 2 x to apply appropriate textual concepts.]