Given that $$I_1=\int_0^1x^2f(x)dx=\frac {1}{3}$$ and $$I_2=\int_0^1(f'(x))^2dx=7$$ Find the value of $$J=\int_0^1f(x)dx$$ In the first equation using a bit of symmetry I substituted $x=1-t$ and after solving I reached $$I_1=2\int_0^1xf(x)dx +\int_0^1 x^2f(1-x)dx-J$$
I do not know how to introduce the $I_2$ term in the equation. I tried integration by parts but it just made things messier.