How To Find the Integral of e When the Exponent Has Multiple Variables

Question

My calculus class just started going over "Separation of Variables" where you have something like $\frac{dy}{dx} = x$ and have to solve for $y$.

I understand that to do this, one must get dx on one side and dy on the other, and then take the integral of both sides. Tonight I was assigned some homework problems, one of which looked like this:

$\frac{dr}{ds} = e^{r-2s}$ I turned this into: $r = \int e^{r-2s} ds$

However once I get to this step I am unsure of how to go further. I was thinking I would turn it into this:

$-\frac{1}{2} \int 2e^{r-2s} ds$

But I don't know how to find the integral of the e considering the fact that there are two variables in the exponent. Can anyone give me any info/tips on this topic?

Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. — Wolgwang, Dec 21 '21 at 03:03
Oh ok, thank you for the tip. I just joined this so I am still figuring things out. — Jack, Dec 21 '21 at 03:13
I just edited my post to give more information, does it look better know? — Jack, Dec 21 '21 at 03:17
$$\mathrm{e}^{r-2s} = \mathrm{e}^r\mathrm{e}^{-2s} = \frac{\mathrm{e}^r}{\mathrm{e}^{2s}}\text{.}$$ — Eric Towers, Dec 21 '21 at 03:20
@EricTowers this makes sense in terms of breaking it down but I'm still unsure of how to solve the integral. Do I only go through the +1 to the exponent and divide process for the s because it is ds? — Jack, Dec 21 '21 at 03:26
Move everything containing "$s$"s (including exponentials and differentials) to one side and everything containing "$r$"s to the other side. Then integrate. — Eric Towers, Dec 21 '21 at 03:27
@Jack Better...but you should type Mathjax. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — Wolgwang, Dec 21 '21 at 03:30
Oh, I forgot that I could still do that. That makes much more sense. Thank you! — Jack, Dec 21 '21 at 03:32
So the answer to the question literally asked in the title is: you don't. But I think you got a couple of good tips in the comments anyway! — David K, Dec 21 '21 at 03:45
Yeah, I was able to use these tips and then some outside research to figure it out in the end. — Jack, Dec 21 '21 at 03:55
@Jack: you can post an answer to your own question. It might then be helpful to others. — robjohn, Dec 21 '21 at 18:50

How To Find the Integral of e When the Exponent Has Multiple Variables

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