< Pi (mathematical constant) | ProofsRevision as of 05:26, 16 September 2009 by imported>Peter Schmitt
We work out the following integral:
![{\displaystyle I\equiv \int _{0}^{1}{\frac {t^{4}(1-t)^{4}}{1+t^{2}}}\,\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c33b7a45acb8f44016b4aa34a30ad2bab12d0f1e)
One can divide polynomials in a manner that is analogous to long division of decimal numbers. By polynomial division one shows that
![{\displaystyle {\frac {t^{4}(1-t)^{4}}{1+t^{2}}}={\frac {t^{8}-4t^{7}+6t^{6}-4t^{5}+t^{4}}{1+t^{2}}}=t^{6}-4t^{5}+5t^{4}-4t^{2}+4-{\frac {4}{1+t^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/778c0f83b01c971b33557906b6c77c04e4f783ac)
where −4 is the remainder of the polynomial division.
One uses:
![{\displaystyle \int _{0}^{1}t^{n}\,\mathrm {d} t={\frac {1}{n+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b81c94cb71e3b173a6a9f0425d53030baeaa2b8e)
for n=6, 5, 4, 2, and 0 and one obtains
![{\displaystyle \int _{0}^{1}(t^{6}-4t^{5}+5t^{4}-4t^{2}+4)\,\mathrm {d} t={\frac {1}{7}}-{\frac {4}{6}}+{\frac {5}{5}}-{\frac {4}{3}}+{\frac {4}{1}}={\frac {22}{7}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d61143fc02ad75d518dc612ace1c0e6aa590c65c)
The following holds
![{\displaystyle -4\int _{0}^{1}{\frac {1}{1+t^{2}}}\,\mathrm {d} t=-4\left[\arctan(t)\right]_{0}^{1}=-4{\frac {\pi }{4}}=-\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/675e06071155885c59b136aeacb2971bf702bf76)
The latter integral is easily evaluated by making the substitution
![{\displaystyle t=\tan(x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b45cf956abf2f0facd3a0d86390aca76b2f4e01)
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 I is positive. Finally,
![{\displaystyle 0<I={\frac {22}{7}}-\pi \quad \Longrightarrow \quad {\frac {22}{7}}>\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6bc8e90c835e0dd2ae116a9f666d98fff0c9c7)
which was to be proved.