What is the integral of this expression:
$$\int\operatorname{e}^{ts}dt$$ where t is the variable and s is the parameter. i want it in this general form and also maybe slight variations such as if $t$ now becomes $t^2$. but the exact problem I am facing is this: $$\frac{dQ}{dt} + \frac{rQ}{100} = \frac{r}{4}$$Q is a function of time $t$. I want to solve this differential equation. Thanks for any help you can give me(I know it's a simple one but help will be much appreciated)
Hint for the differential equation
I suppose that r is a constant $$\frac{dQ}{dt} + \frac{rQ}{100} = \frac{r}{4}$$ I used the integrating factor $\mu=e^{rt/100}$ but it's also separable $$Q'e^{rt/100} + e^{rt/100}\frac{rQ}{100} = e^{rt/100} \frac{r}{4}$$ $$(Qe^{rt/100})' = e^{rt/100} \frac{r}{4}$$ Integrate $$(Qe^{rt/100}) = \frac{r}{4}\int e^{rt/100}dt $$ $$(Qe^{rt/100}) = \frac{100r}{4r}e^{rt/100} +K\frac{r}{4}$$ $$(Qe^{rt/100}) = 25e^{rt/100} +K\frac{r}{4}$$ $$Q(t)= \frac{r}{4}Ke^{-rt/100}+25$$ $$\boxed{Q(t)= Ce^{-rt/100}+25}$$
Edit
Note that $$\int e^{ax}dx= \frac {e^{ax}} a$$ $$ \implies \int e^{rt/100}dt= {e^{rt/100}} \frac {100}{r}$$