MathDB

Today's calculation of Integral 852

Source:

November 10, 2012

calculusintegrationtrigonometryalgebrapolynomialabsolute valuecalculus computations

Problem Statement

Let

f(x)

be a polynomial. Prove that if

\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)

, then all coefficients of

f(x)

are 0 for each case as follows.
(1)

g_n(x)=(1+x)^n

(2)

g_n(x)=\sin n\pi x

(3)

g_n(x)=e^{nx}