derivada de integral

Teorema:
Si ( F(x) = ∫ [0 --> sin(x)]-[ f(x) ] d[x] & f(x) és creishent ) ==> d_{x}[F(x)] [< f(x)


Demostració:
d_{x}[F(x)] = f(sin(x))·d_{x}[sin(x)] = f(sin(x))·cos(x) [< f(sin(x)) [< f(x)


Teorema:
Si ( F(x) = ∫ [0 --> x^{n+1}]-[ f(x) ] d[x] & f(x) és creishent ) ==> ...
...[∀x][ 0 [< x [< 1 ==> d_{x}[F(x)] [< (n+1)·f(x) ]


Demostració:
d_{x}[F(x)] = f(x^{n+1})·(n+1)·x^{n} [< (n+1)·f(x^{n+1}) [< (n+1)·f(x)