( e^ 2 x [ (3 x+4)+5 x^2 ] d x= )

Step-by-step solution

( aligned & e^ 2 x [ (3 x+4)+5 x^2 ] d x & = e^ 2 x (3 x+4) d x+5 e^ 2 x x^2 d x & [ e^ a x (b x+c) d x= e^ a x a^2+b^2 (a (b x+c)+b (b x+c))+c ] aligned ) ( aligned & =e^ 2 x [ 2 13 (3 x+4)+ 3 13 (3 x+4) ]+ 5 x^2 2 e^ 2 x & =e^ 2 x ( 2 13 (3 x+4)+ 3 13 (3 x+4) ) + 5 x^2 2 e^ 2 x - 5 2 x e^ 2 x + 5 2 e^ 2 x d x aligned ) ( aligned & =e^ 2 x [ 2 13 (3 x+4)+ 3 13 (3 x+4) ] + 5 2 x^2 e^ 2 x - 5 2 x e^ 2 x + 5 4 e^ 2 x & =e^ 2 x [ 2 13 (3 x+4)+ 3 13 (3 x+4)+ 5 x^2 2 - 5 2 x+ 5 4 ] +c aligned ) Hence, option (a) is corr