Q:

Explicit sequence of polynomials which converge to zero but have non-zero integral

I'm looking for a sequence of polynomials on $[0,1]$ which converge to zero but have non-zero integral.
I previously solved this problem using a common example of a sequence of continuous functions $s_n$ with this property (namely the one which "squeezes" a triangle to the y-axis) and then arguing that stone weierstrass, applied to each $s_n$, gives a sequence of polynomials $p_{m,n}$ and then you can find a diagonal sequence which converges to zero and has integral converging to a non-zero value.
However, all of this would be a lot cleaner if I just had an explicit example of a sequence of polynomials which have this property. Can anyone share? I don't want a piece-wise definition - this sequence of triangles can be defined piecewisely as three polynomials (two linear ones and a zero one).

A:

Try $P_n(x) = (n+1)(n+2) x^n (1-x)$.

