I want to find the formula for the Gauss quadrature that integrate the Integral $\int_{-1}^1f(x)\sqrt{|x|}\, dx$ exactly for every cubic polynomial $f$.
What exactly do we have to do here?
Is $\sqrt{|x|}$ the weighting function?
Could you explain me all the procedures step by step?
Since the integration rule is to be exact for polynomials of degree three, you will get four conditions that can set four parameters. So, we want a rule of the form $Q(f) = w_0 f(x_0) + w_1 f(x_1)$, such that $$ Q(1) = I(1), Q(x) = I(x), Q(x^2)=I(x^2), Q(x^3)=I(x^3). $$
This leads to the system $$ \begin{cases} w_0+w_1 = \frac 43\\ w_0 x_0 + w_1 x_1 = 0\\ w_0 x_0^2 + w_1 x_1^2=\frac 47\\ w_0x_0^3+w_1 x_1^3 = 0 \end{cases} $$
Now you just need to solve for $x_0, x_1, w_0, w_1$, which is not hard in this case, to get $$ Q/f)= \frac 23 f\left(-\sqrt{\frac 37}\right)+\frac 23 f\left(\sqrt{\frac 37}\right). $$