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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?

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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 $a_1, a_2, x_1, x_2$.

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