%%% Example of how the symbolic toolbox 
%%% can save you time evaluating integrals.
syms t;
int('t^2*exp(t)',-1,1) %definite integral on [-1,1]


%%% computing f_3 = proj_W(f) when 
%%% W = space of all polynomials of degree <= 2 defined on [-1,1]
%%% and f(x) = exp(x)
G = [ 2   0   2/3
      0  2/3   0
     2/3  0   2/5]; %%% Gram matrix for non-orthogonal basis 
                    %%% {1, x, x^2} (computed previously by hand,
                    %%% or with symbolic toolbox)


syms e  %%% choose this if you want an exact expression of c
%e = exp(1); %%% choose this if you want plots
b = [e - 1/e
      2/e
     e - 5/e]; %%%  b(i) = (f,phi_i), where phi_i(x) = x^(i-1)

c = G\b
x = linspace(-1,1,200)';
f3 = c(1)*ones(size(x)) + c(2)*x + c(3)*x.^2;

plot(x,[exp(x),f3])