Let `f:[1,oo]` be a differentiable function such that `f(1)=2.` If `6int_1^xf(t)dt=3xf(x)-x^3` for all `xgeq1,` then the value of `f(2)` is

Explanation

`6int_(t)^(x)f(t)dt = 3xf(x)-x^(3)`
Differentiating w.r.t.x, we get
or `6f(x)=3f(x)+3xf^(')(x)-3x^(2)`
or `x(dy)/(dx) -y=x^(2)`
or `(xdy-ydx)/(x^(2))=dx`
or `int(xdy-ydx)/x^(2)=intdx`
or `y/x =x+c`
Given, `f(1)=2`
or `c=1`
or `y=x^(2)+x`
Note if we put `x=1` in the given equations, we get `f(1)=1//3`.