Question

If f (x) is differentiable and ∫ x f (x)dx = 2/5 t^5, where x∈ [0,t^2] then f (4/25) equals

Answer 1

Answer:

2/5

Step-by-step explanation:

First we will be differentiate both sides

                 ( ∫ x f(x) dx ) ' =   (2/5 t^5)' Equation

1.   ( ∫ x f(x) dx ) '

              x= t²

              dx= 2t

Differentiating the integration will cancel out the integration sign and the function inside it will be left

         

              ( ∫ x f(x) )' = x f(x)

         so  ( ∫ x f(x) dx ) ' = ( x f(x) ) 2t

     2.     (2/5 t^5)'

                5 *( 2/5 t^4)

            =  2t^4

     3.    put x = t² in equation

               t² f(t²) *  2t = 2t^4

               t²f(t²)  =  2t^4 / 2t

               t²f(t²) = t³

               f(t²) =t³/t²

               f(t²)=t

So from here we can derive the relation that f(t) = √t

hence, f(4/25) = √4/25 = 2/5 Ans.