Help with solving $\int_0^n \exp(-rt)\exp\left(\frac sre^{-rt}\right).dt$

Question

Given $$\int_0^n \exp(-rt)\exp\left(\frac sre^{-rt}\right).dt$$

Can you please show the step(s) involved to reach this next line in the textbook:

$$\left[-\frac 1s\exp\left(\frac sr e^{-rt} \right) \right]_{t=0}^{t=n}$$

Is it done (or can it be done) with integration by parts? It's the third $e$ within the second $e$ that completely throws me on how to approach it, any tips on how to deal with that would be really useful, thanks.

Answer 1

Notice $$ \frac{d( e^{-rt} )}{-r} = e^{-rt} dt \implies \frac{s}{r} \frac{d( \frac{r}{s}e^{-rt} )}{-r} = e^{-rt} dt$$

$$ \therefore \int e^{-rt}e^{\frac{s}{r}e^{-rt}}dt = \frac{-s}{r^2}\int e^{\frac{s}{r}e^{-rt}} d( \frac{r}{s}e^{-rt} ) = \frac{-s}{r^2} e^{\frac{s}{r}e^{-rt}} + C = F(t)$$

$$ \therefore F(n) - F(0) = \frac{-s}{r^2} e^{\frac{s}{r}e^{-rn}} - \frac{-s}{r^2} e^{\frac{s}{r}} $$