TITLE: Definite integral of product of functions
QUESTION [6 upvotes]: I know it's not correct to write:
$$\int_{a}^{b}f(x)g(x) dx = \int_{a}^{b}f(x)dx\int_{a}^{b}g(x)dx$$
This result seems obvious, but I can't think of a way to prove that $\int_{a}^{b}f(x)g(x) dx$ can't be expressed as a function of the form :$$F\left(\int_{a}^{b}f(x)dx;\int_{a}^{b}g(x)dx\right)$$
Is it something simple I'm missing ? How can one prove that a definite integral can't be represented as mentioned before ?

REPLY [2 votes]: Hint:
Let:
$$
\int_a^x f(t)dt=F(x) \qquad  \int_a^x g(t)dt=G(x) 
$$
From the fundamental  theorem of calculus we have
$$
\dfrac{d}{dx} F(x)=f(x) \qquad \dfrac{d}{dx} G(x)=g(x) 
$$
and, using the product rule for derivative:
$$
\dfrac{d}{dx}\left[\int_a^x f(t)dt\int_a^x g(t)dt\right]=\dfrac{d}{dx}\left[F(x)G(x)\right]=G(x)f(x)+F(x)g(x) \ne f(x)g(x)
$$