If f(x)=Integral (t^3+6t^2+4)dt from 0 to x
then f''(x)=?
what's the problem? The integral is just normal power stuff
∫ t^3 + 6t^2 + 4 dt
= 1/4 t^4 + 2t^3 + 4t
now evaluate that at x and 0, and you have
f(x) = 1/4 x^4 + 2x^3 + 4x
so, f"(x) = 3x^2 + 12x