< Pi (mathematical constant) | ProofsRevision as of 05:55, 16 September 2009 by imported>Paul Wormer
We work out the following integral:
It is possible to divide polynomials in a manner that is analogous to long division of decimal numbers. By doing this it, can be shown that
where −4 is the remainder of the polynomial division.
Using:
for n=6, 5, 4, 2, and 0, respectively, one obtains
The following holds
The latter integral is easily evaluated by making the substitution
The integrand (expression under the integral) of the integral I is everywhere positive on the integration interval [0, 1] and, remembering that an integral can be defined as a sum of integrand values, it follows that the integral is positive. Finally,
which was to be proved.