Integrated rate laws
From the reaction 

$$\ce{2A -&gt; B}$$

we have $a = 2$, which leads to the differential equation:

$$\frac{d[A]}{dt} = - k [A]^2$$

which then leads to the integrated rate law you've cited.  The $2$ is nowhere as a coefficient, it is related to the order of the equation, and determines which integral/form should be used.  Similar conventions also apply for 1st and zeroth order kinetics.  I disagree with your instructor.

It should be stressed that these rate laws apply to elementary reactions, not complex reactions.  Experimental measurement may differ from that predicted using an elementary approach, which then leads to the conclusion there are other steps involved.  In the case of OP's comment regarding oxidation of carbon monoxide with NO2, elementary reactions would predict a rate proportional to $$k [\ce{NO2}][\ce{CO}]$$ if the reaction is the result of a simple bimolecular collision. The fact that a different rate is observed indicates this process is more elaborate.