# Gauss Integration of $\sqrt(x)$

I want to construct a gauss integration for the weight function $$w(x) = x^{1/2}$$ for

$$\int_{0}^{1}x^{1/2}f(x)dx = a_{1}f(x_{1})+a_{2}f(x_{2})$$

Solving

\begin{align*} a_{1}+a_{2} =& \int_{0}^{1}x^{1/2} dx = \frac{2}{3} \\ a_{1}x_{1}+a_{2}x_{2}=&\int_{0}^{1}x^{1/2}x dx =\frac{2}{5}\\ \end{align*}

Solving this system will yield $$a_{1},a_{2}$$? and then what?

Quadrature points $$x_1,x_2$$ are also unknown. You need two more equations corresponding to $$x^2$$ and $$x^3$$. Then you solve for $$a_1, a_2, x_1, x_2$$.