Prove that a regular curve $\alpha$, with non vanishing curvature, has constant torsion $\tau = \cfrac{1}{a}, a \neq 0$ if, and only if: $$\alpha(t) = a\left( \int f_1(t) \ dt,\int f_2(t) \ dt, \int f_3(t) \ dt \right)$$
where $(f_1, f_2,f_3) = F \times F'$ and $F$ is a vector valued function such that $||F(t)|| = 1$ and $F \cdot(F' \times F'') \neq 0$.
I don't know how to start solving this. My only idea was to try solving the differential equations of the TNB frame, but I have too little information. I'd appreciate any help.
You dont have to solve them just read them. The condition you give is that $$\alpha^{\prime}(t)=aF\times F^{\prime}$$ So you just need to identify $F$. And if you assume that $\alpha$ is parameterized by arc length, $\mathbf{t}=\alpha^{\prime}$. And it seems that $F=\mathbf{b}$. I leave the rest of the verification up to you.