My brother brought me this same question, also on this website.
$$\int_{0}^{x}f(t)dt = x+\int_{x}^{1}tf(t)dt \tag{1}$$
When I try to solve it, I also did same thing. Differentiation leads to
$$f(x) = 1-xf(x)\\ f(x) = \frac{1}{1+x} \\ \int_{0}^{1}f(x) dx = \ln 2 $$
But try putting $x = 1$ in original equation $(1)$,
$$\int_{0}^{1} f(t) dt = 1$$
so what has happened? We get two contradictory results. Is it still true that $f(1) = 1/2$? Where is the mistake?