Integrating of ODE

Question

Let's assume that I write change in value of differentiable function $f$ over a time interval at $t$ of infinitesimal length $dt$ as $$df_t = t dt$$ So I basically get simple ODE. The solution I see in a book is to simple integrate that eqaution and then I get the solution $f_t=\frac{1}{2} t^2$. My question is: how is this integration formally made? As far as I know, when I integrate a side of equation, it means that I put in into this expression: $ \int ... dt$. However, I already have in the both side of equations with different variables of integration ($df_t$ and $dt$). So if I followed above logic, I would have $$ \int df_t\ dt = \int tdt \ dt$$ which doesn't have any sense. Thank you in advance for your help.

Answer 1

You should realize that

$$ \int {\rm d}\mu = \mu + c \tag{1} $$

where $c$ is a constant. Once you agree with this statement, the rest follows immediately, for example, note that

$$ {\rm d}\left(\frac{1}{2}t^2\right) = t~{\rm d}t \tag{2} $$

So your original equation becomes

$$ {\rm d}f_t \stackrel{(2)}{=} {\rm d}\left(\frac{1}{2}t^2\right) \tag{3} $$

Integrating at both sides you get

\begin{eqnarray} \int {\rm d}f_t &=& \int {\rm d}\left(\frac{1}{2}t^2\right) \\ f_t &\stackrel{(1)}{=}& \frac{1}{2}t^2 + c \tag{4} \end{eqnarray}