First order reaction

I am little bit confuse about relating the half life time period of a first order reaction something like


$$\ce{2A -> B + C}$$


Here the rate of reaction will be half of the rate at which $\ce{A}$ is used up.


$$\text{Rate of reaction} = k[\ce{A}]$$


$$\text{Rate of disapperance of}~\ce{A} = 2k[\ce{A}]$$


Integrating it half life will be $\displaystyle\frac{\ln 2}{2k}$.
Am I wrong somewhere? 


