∫0xf(t)dt=x+∫x1tf(t)dt (given);
When x=0, ∫0xf(t)dt=0, so ∫01tf(t)dt=0 (see also later);
∫x1tf(t)dt=∫0xf(t)dt-x;
x=∫0xdt;
∫x1tf(t)dt=∫0xf(t)dt-x=∫0x(f(t)-1)dt;
∫x1tf(t)dt=(1-x)(∫x1f(t)dt)-∫x1f(t)dt (integration by parts);
∫01tf(t)dt=∫01f(t)dt-∫01f(t)dt=0;
∫01tf(t)dt=∫01tf(t)dt+∫01(f(t)-1)dt=∫01(tf(t)+f(t)-1)dt=0 and, since ∫01tf(t)dt=0, ∫01(f(t)-1)dt=0.
f(t)-1=0, so f(t)=1, therefore f(1)=1.