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## 'Mechanical Integration' printed from http://nrich.maths.org/

There is a theory of polynomials which enables you to find the
integral of a polynomial by simply evaluating it at some special
points and adding certain multiples $\Lambda_1, \Lambda_2,\ldots$
of these values. For example, for all polynomials $q$ of degree
less than six, the special points are $-\sqrt{3/5}$, $0$ and
$+\sqrt{3/5}$, and $$\int_{-1}^1 q(x)dx = \Lambda_1q(-\sqrt{3/5}) +
\Lambda_2q(0) + \Lambda_3q(+\sqrt{3/5}).\quad (1)$$ Find the
multiples $\Lambda_1,\ \Lambda_2$ and $\Lambda_3$ by considering
the three polynomials $q(x) = 1$, $q(x) = x$ and $q(x) =
x^2$.

With these values of $\Lambda_1,\ \Lambda_2$ and $\Lambda_3$ show
that the mechanical integration given by equation (1), which uses
the values of the polynomial at the three special points, gives the
value of the integral of ALL quadratic polynomials.

Now go on to show that the same formula gives the integral of ALL
cubic, quartic and quintic polynomials.

Does the formula (1) hold for $q(x)=x^6$?