For positive t , let , be the root of the quadratic equation x^2+t^2 x-2 t=0 . If minimum value of _ -1 ^2 ( (x+ 1 ^2 ) (x+ 1 ^2 + 1 ) ) d x is p +q where p is a rational number and q N , then the value of [p]+q is (where [.] denotes G.I.F)

Step-by-step solution

(x+ 1 ^2 ) (x+ 1 ^2 )+ 1 =x^2+ ( ^2+ ^2 ( )^2 ) x+ 1 ( )^2 + 1 =x^2+ ( t^4+4 t 4 t^2 ) x+ 1 4 t^2 - 1 2 t Let I= _ -1 ^2 x^2 d x+ 3 2 ( t^2 4 + 1 t )+3 ( 1 4 t^2 - 1 2 t ) d ~d dt = 3 2 ( t 2 - 1 t ^2 - 1 t ^3 + 1 t ^2 )=0 t ^4=2 t= (2)^ 1 / 4 Min I(t)=3+ 3 2 ( 2 4 + 1 2^ 1 / 4 + 1 2 2 - 1 2^ 1 / 4 )=3+ 3 2 2 2 =3+ 9 8 [ 9 8 ]+3=1+3=4