producte integral exponencial

f(x) [v(x)v] g(x) = ∫ [ D_{x}[f(x)]·D_{x}[g(x)] ] D[x]


( f(x) [v(x)v] g(x) ) [v(x)v] h(x) = f(x) [v(x)v] ( g(x) [v(x)v] h(x) )
∫ [ ( D_{x}[f(x)]·D_{x}[g(x)] )·D_{x}[h(x)] ] D[x] = ∫ [ D_{x}[f(x)]·( D_{x}[g(x)]·D_{x}[h(x)] ) ] D[x]


e^{x} [v(x)v] g(x) = ∫ [ D_{x}[g(x)] ] D[x] = g(x)


f(x) [v(x)v] ( f(x) )^{[v(x)v](-1)} = e^{x}


teorema:
e^{f(x) [o(x)o] g(x)} = e^{f(x)} [v(x)v] e^{g(x)}


demostració:
D_{x}[ e^{f(x) [o(x)o] g(x)} ] = D_{x}[ e^{f(x)} [v(x)v] e^{g(x)} ]


proposició:
e^{x [o(x)o] f(x)} = e^{x} [v(x)v] e^{f(x)}