Calculate some values for some even and odd functions with given properties

Let $f$ be an odd function, integrable everywhere, with

$f(5) = 7, \qquad f(0) = 0.$

Let $g$ be an even function, integrable everywhere, with

$g(x) = f(x+5), \qquad f(x) = \int_0^x g(t) \, dt \qquad \text{for all } x.$

Prove the following:

Proof. We compute using the given properties,
$\begin{align*} f(x+5) = g(x) && \implies && g(-x) &= f(-x + 5) \\ && \implies && -g(x) &= f(x-5) & (g \text{ is odd}, f \text{ is even}). \ \blacksquare \end{align*}$
Again, we compute using the given properties of $f$ and $g$ ,
$\begin{align*} \int_0^5 f(t) \, dt &= \int_0^5 g(t-5) \, dt \\ &= \int_{-5}^0 g(t) \, dt & (\text{Translation})\\ &= \int_5^0 g(-t) \, dt \\ &= - \int_0^5 g(-t) \, dt \\ &= \int_0^5 g(t) \, dt & (g \text{ is odd}) \\ &= f(5) = 7. \qquad \blacksquare \end{align*}$
Finally,
$\begin{align*} \int_0^x f(t) \, dt &= \int_0^x g(t-5) \, dt \\ &= \int_{-5}^{x-5} g(t) \, dt \\ &= \int_{-5}^0 g(t) \, dt + \int_0^{x-5} g(t) \, dt\\ &= f(5) + f(x-5) \\ &= g(0) - g(x). \qquad \blacksquare \end{align*}$

Stumbling Robot