I'm really lost here. I'm supposed to show that this transfer frunction for a first order system: $$G(s) = \frac{k}{\tau s+1}$$
with a pole placement at
$$s = -\beta\pm i\beta$$
corresponds to a PI-controller with parameters:
$$K = \frac{2\beta \tau -1}{k}$$ $$T_I=\frac{Kk}{2\beta^2 \tau}$$ Now, I have tried using $1+G(s)F(s)=0$ for a closed-loop system, and $F(s) = K(1+\frac{1}{sT_I})$ for a PI-controller, since those are my given conditions. But when I use that and my given $G(s)$, it just gets messy and I get an answer with an imaginary part.
Hint:
$$1+F(s)G(s)=0 \implies \tau s^2+(1+kK)s+\frac{kK}{T_I}=0$$
Note that $kK= 2\beta \tau-1$ and $kK/T_I=2\beta^2\tau.$ Using these and assuming $\tau\neq 0$ we get:
$$\tau s^2+2\beta\tau s+2\beta^2\tau=0 \implies s^2+2\beta s+2\beta^2=0.$$
This quadratic equation has the roots:
$$s_{1,2}=-\beta\pm j\sqrt{\beta}.$$