Let f : R R be a twice differentiable function such that f(2)=1 . If F (x)=x f(x) for all x R , ₀^2 x ~F ^ (x) d x=6 and ₀^2 x^2 ~F ^ (x) d x=40 , then F ^ (2)+ ₀^2 ~F (x) d x is equal to :

Step-by-step solution

aligned & ₀^2 xF ^ ( x ) dx =6 & = . xF ( x ) |₀ ^2- ₀^2 f ( x ) dx =6 & =2 ~F (2)- ₀^2 xF ( x ) dx =6[ f (2)=2 ~F (2)=2] & ₀^2 xF ( x ) dx =-2 ...(1) & ₀^2 ~F ( x ) dx =-2 ...(2) aligned Also ₀^2 x^2 F^ (x) d x= .x^2 F^ (x) |₀ ^2-2 ₀^2 x^ F^ (x) d x=40 aligned & =4 F^ (2)-2 6=40 & F^ (2)=13 & F^ (2)+ ₀^2 F(x)=13-2=11 aligned