Given the Euclidean vector space
\left(\mathcal P_{\leq 2}, \langle \ , \ \rangle \right) with
\displaystyle \langle p, q \rangle = \int_{-1}^{1} f(x)g(x) \; dx.
Compute the SCP for {\color{red}p}, \ {\color{blue}q} \; : [-1,1] \to \mathbb R
with {\color{red}p(x) = Ax^2 + Bx +C} und
{\color{blue}q(x) = Dx + 1}.
\langle p, q \rangle =
(2*(B*D+A)+6*C)/3
Compute the SCP, i.e. the integral
\displaystyle \langle p, q \rangle =
\int_{-1}^{1} {\color{red}p(x)}\ {\color{blue}q(x)} \; dx
with the fundamental theorem of Calculus.
This becomes \displaystyle
\int_{-1}^{1} {\color{red}\left(Ax^2 + Bx +C\right)}\
{\color{blue}(Dx + 1)} \; dx =
\left(negParens(fractionReduce(A*D,4))x^4 + fractionReduce(B*D+A,3)x^3 +
fractionReduce(C*D+B,2)x^2 + Cx \right)\biggl|_{-1}^{1} =
fractionReduce(2*(B*D+A)+6*C,3).