Past exam of the approximation theory and finite element methods course of the University of Oxford / 2009 / A / 2 / c

The monic degree-zero polynomial is ${\varphi }_0=1$. Orthogonality of $x-a$ to one gives $a={\int }_0^{1}xdx=1/2$, so ${\varphi }_1=x-1/2$.

The relevant integrals areInsert these into the three-term recurrence for monic orthogonal polynomials:ThusTheir mutual orthogonality can also be checked directly by integrating on $[0,1]$.