While solving some exercises on orthnormal set on Hilbert spaces, I tried to solve the following problem: Let $\left\{ f_{1},f_{2},...,f_{9}\right\} $ be an orthonormal set in $L^{2} \left[ 0,1\right] .$ Assume that $$ \int_{0}^{1}6xf_{1}\left( x\right) dx=2 $$ and $$ \int_{0}^{1}6xf_{2}\left( x\right) dx=2\sqrt{2}. $$ What can you say about the value of $\int_{0}^{1}6xf_{5}\left( x\right) dx?$ Give reasons.
I tried to write the function $6x$ as a linear combination of this orthonormal set and made use of the given values but in vain. Any help with that is very appreciated.