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852

Part of 2012 Today's Calculation Of Integral

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Today's calculation of Integral 852

Source:

11/10/2012
Let f(x)f(x) be a polynomial. Prove that if ∫01f(x)gn(x) dx=0 (n=0, 1, 2, ⋯ )\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots), then all coefficients of f(x)f(x) are 0 for each case as follows.
(1) gn(x)=(1+x)ng_n(x)=(1+x)^n
(2) gn(x)=sin⁡nπxg_n(x)=\sin n\pi x
(3) gn(x)=enxg_n(x)=e^{nx}
calculusintegrationtrigonometryalgebrapolynomialabsolute valuecalculus computations