Using an integrating factor to solve an ODE I have been simply asked to integrate the following:
$dc/dt = 1-c-a c$.
I have used the integrating factor as $e^{(1+a) t}$.
Then finish with $c=1/(1+a)$, however my lecturer gets $c=\frac{1}{1+a}(1-e^{(a+1)t})$.
Could anyone at all show me the method to get this. I know it is meant to be a relatively easy question, but  i just don't know  what i am doing wrong. 
Thank you.

$$
\frac{dc}{dt}+(1+a)c = 1
$$
multiply by integrating factor $e^{(1+a)t}$:
$$\begin{align}
&e^{(1+a)t}\frac{dc}{dt}+(1+a)e^{(1+a)t}c=e^{(1+a)t}
\\
&\frac{d}{dt}\left(e^{(1+a)t}c\right)=e^{(1+a)t}
\\
&e^{(1+a)t}c = \int e^{(1+a)t}\,dt = \frac{e^{(1+a)t}}{1+a}+A
\\
&c = \frac{1}{1+a}+Ae^{-(1+a)t}
\end{align}$$
for some constant $A$.  Presumably there is an initial condition (that R.M. has not told us) to let the lecturer determine the constant $A$
