Given that d d x [ ₀^ (x) f(t) d t ]= ^ (f(x)) f^ (x) . If ₀^ 3^3 f(t) d t=x^2 2 x , then the value of f(8) is

Step-by-step solution

₀^ x^3 f(t) d t=x^2 2 x Differentiating both sides, we get d d x [ ₀^ x^3 f(t) d t ]= d d x [x^2 2 x ] aligned & f (x^3 ) 3 x^2=2 x (2 x)+2 x^2 (2 x) & f (x^3 )= 2 x 3 x^2 [ (2 x)+x (2 x)] & f (x^3 )= 2 3 x [ (2 x)+x (2 x)] aligned Put x=2 : aligned & f(8)= 2 3 2 [ (4 )+2 (4 )] & = 1 3 [0+2 1] f(8)= 2 3 . aligned