TITLE: $1^3 + \dotsb + n^3 = (1 + \dotsb + n)^2$: reason?
QUESTION [6 upvotes]: We have 
$$
1^3 + \dotsb + n^3 = (1 + \dotsb + n)^2
$$
as we can establish by induction. But why does this hold? Can we connect it to something else?

REPLY [4 votes]: Funnily, we also have
$$\int_0^x t^3\: dt = \left(\int_0^x t\: dt\right)^2.$$