Let `f:[1,oo)` → `[2, ∞)` 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

`6 int_1^x f(t)dx= 3xf(x)-x^3`

differentiate wrt x``

`6f(x) = 3f(x) + 3xf'(x) - 3x^2`

`cancel(3)f(x) = cancel(3)xf'(x) - cancel(3)x^2`

`y = x(dy/dx) - x^2`

`ydx-xdy= -x^2dx`

now`(xdy-ydx)/x^2=dx`

`=> intd(y/x) = intdx`

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