Given the weight function $w(t) \equiv 1$, for $t\in [0,1]$ build the Gaussian quadrature formula of order $2$.
The formula in this case has the form
$$ \int_0^1 f(\tau)d\tau \approx b f(c) $$
For this, I find an orthogonal polynomial $1-2t$, whose root is $c:=\frac{1}{2}$. To find $b$, I need to take the interpolated polynomial $$ t-\frac{1}{2} $$
if I'm not mistaken, since
$$p_j(t) = \prod\limits_{k=1, k\ne j}^\nu\frac{t-c_k}{c_j-c_k}$$
where $j=1,...,\nu$. In this case we have just one $c_k$, so it looks like we can just take $t-\frac{1}{2}$.
Then $$ b = \int_0^1 (\tau -\frac{1}{2})d\tau$$ which is just $0$. So something I'm doing is not right. I think I'm erring on the interpolating polynomial side. Would appreciate some help.