The integral of $\frac{1}{\tau_1-\tau_2}(e^{-t/\tau_1}-e^{-t/\tau_2})$

Question

If we have

$$ \frac{dx}{dt}=\frac{1}{\tau_1-\tau_2}(e^{-t/\tau_1}-e^{-t/\tau_2}), $$

what is $x$, and what are the steps by which one comes to that solution?

Answer 1

$$\begin{align*} \int \frac{1}{\tau_1 - \tau_2}(e^{-t/\tau_1} - e^{-t/\tau_2}) dt &= \frac{1}{\tau_1 - \tau_2}\int e^{-t/\tau_1} dt - \frac{1}{\tau_1 - \tau_2}\int e^{-t/\tau_2} dt \\ &= \frac{-\tau_1}{\tau_1 - \tau_2} e^{-t/\tau_1} - \frac{-\tau_2}{\tau_1 - \tau_2} e^{-t/\tau_2} + C \end{align*}$$

Since

$$ \int e^{-t/\tau_1} dt = -\tau_1 e^{-t/\tau_1} + C$$