A kind of polynomial: $\int_0^1 P^2 = (n+1)(\int_0^1 P)^2$

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A kind of polynomial: $\int_0^1 P^2 = (n+1)(\int_0^1 P)^2$

Postby elim » Mon Mar 05, 2012 8:49 am

Prove that
$\displaystyle{ \left(\int_0^1 x^k P(x) dx = 0,\ (1\le k\le n = \deg(P))\right)}$
$\displaystyle{ \implies \left(\int_0^1 (P(x))^2dx = (n+1)\bigg(\int_0^1 P(x)dx \bigg )^2 \right ) }$

http://www.artofproblemsolving.com/Foru ... 7&t=467693
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