Let $V = \mathcal{C}[-1, 1]$ be the real vector space of real-valued continuous functions defined on the closed interval $[-1, 1]$. $V$ is an inner product space with the inner product $\langle f, g\rangle = \int_{-1}^{1} f(x)g(x) dx$.
Find the least square approximation to $p = x^{1/3}$ in $W = \textrm{span} \left\{q_o = 1, q_1=x, q_2 = x^2 - \frac{1}{3} \right \}$.
I thought about using $\textrm{proj}_W p$, but I ran into the trouble of evaluating $\int_{-1}^{1} x^{1/3} dx$.
Any ideas?