If 2 x^2+5 x+9 x^2+x+1 ~d x=x x^2+x+1 + x^2+x+1 + _e |x+ 1 2 + x^2+x+1 |+ C , where C is the constant of integration, then +2 is equal to _______.

Step-by-step solution

aligned & 2 x ^2+5 x +9= A ( x ^2+ x +1 )+ B (2 x +1)+ C & A =2 ~B = 3 2 C = 11 2 & 2 x ^2+ x +1 dx + 3 2 2 x +1 x ^2+ x +1 dx + 11 2 dx x ^2+ x +1 & 2 ( x + 1 2 )^2+ ( 3 2 )^2 dx +3 x ^2+ x +1 + 11 2 dx ( x + 1 2 )^2+ ( 3 2 )^2 & 2 ( x + 1 2 2 x ^2+ x +1 + 3 8 ( x + 1 2 + x ^2+ x +1 ) )+3 x ^2+ x +1 & + 11 2 n ( x + 1 2 + x ^2+ x +1 )+ C & = 7 2 = 25 4 & +2 =16 & aligned