Integrating linearly I came across this question and just want to make sure my understanding is correct. 
I need to find the general solution of:
$$
\frac{dx}{dt} = a(1 - x)
$$
In this case, I'm finding the how $x$ changes with respect to $t$ so I'm integrating with respect to $t$. Does that mean the answer is $at - xat + C$? 
Thanks :) 

If you are given, for $x=x(t)$
$$\frac{dx}{dt} = a(1-x)$$
Then, one can re-arrange (by abuse of notation), giving
\begin{align}
\frac{dx}{1-x} &= a\, dt \\
\implies \int \left(\frac{1}{1-x}\right)dx &= a \int dt
\end{align}
The right hand side is straightforward for you, the left requires some knowledge of standard integrals.
Does this help?
