I need a non trivial function $f(x)$ such that $$\int_0^1 x^n f(x) dx=a_n\pi+b_n$$ where $a_n,b_n\in\mathbb{Z}$ and $n\in\mathbb{N}$
We know that $$\pi=4\int_0^1 \sqrt{1-x^2}\ dx$$
By Binomial theorem for rational index $$\pi= 4\int_0^1 \sum_{r=0}^\infty\binom{1/2}{r} x^{2r}\ dx$$
Any help would be highly appreciated.
Edit
If $f(x)=\sin^{-1} x$ then see here $$\int_0^1 x^n \sin^{-1} x \ dx= c_n \pi +d_n$$ but $c_n,d_n\in\mathbb{Q}$