The question in the title is the following:
If $f : \mathbb{R}^n \to \mathbb{R}$ is differentiable and $f(0) = 0$, prove that there exist $g_i : \mathbb{R}^n \to \mathbb{R}$ such that $$f(x) = \sum_{i = 1}^{n} x_i g_{i}(x).$$
And the hint provided is the following: If $h_x(t) = f(tx)$, then $f(x) = \int_{0}^{1} h_x^{'}(t)dt.$
I followed the hint and got the result below, $$f(x) = \int_{0}^{1} \sum_{i = 1}^{n}x_i D_if(tx)dt.$$
Now I'd like to bring the sum outside due to linearity of the integral, but for that I need to show that the functions $h_i : \mathbb{R} \to \mathbb{R}$ defined by $h_i(t) = D_i f(tx)$ are integrable. How do I show this?
I know that there are differentiable real valued functions whose derivatives are not integrable, so I don't know if this result is true at all (by this I mean I think it's possible to slightly tweak such a function with non integrable derivative to get a counterexample). Am I right in thinking so? All the solutions I can find to this problem (for example, here: http://vision.caltech.edu/~kchalupk/spivak.html) just expand the integral without considering integrability of the $h_i$s so I might be missing something basic here.
Any help is appreciated.