Stuck with derivation of the integrated rate equation for a pseudo first order equilibrium reaction
You have made no mistake. Your solution and the solution from the paper are practically identical. The solution from the paper is

\begin{align}
  F(t)=c_1 \frac{1−e^{−c_2 t}}{c_2}=\frac{c_1}{c_2} - \frac{c_1}{c_2} e^{−c_2 t}
\end{align}

which is exactly the form your equation has, i.e. 

\begin{align}
  F(t)=\frac{c_1}{c_2} + K e^{−c_2 t} \qquad \text{with} \qquad K = - \frac{c_1}{c_2} \ .
\end{align}

The integration constant $K$ is not defined the way it is but fixed via the boundary conditions of the reaction. In this case you get $K = - \frac{c_1}{c_2}$ by requiring that at the start of the reaction, i.e. $t=0$, the reactants have not yet reacted with each other and there is no product $\ce{AB}$ present initially, i.e. $F(t\!=\!0) = 0$. This leads to the desired result:

\begin{align}
  F(t\!=\!0) \overset{!}{=} 0 &amp;=\frac{c_1}{c_2} + K e^{0} \qquad \Rightarrow \quad K=- \frac{c_1}{c_2} \ .
\end{align}