Why is the answer to this exponential integral ye^(x/y)dy and not e^(x/y)/y dy?

[steelphantom]

Ok, this isn't a particularly hard integral, but for some reason I don't understand why the answer is what it is. Here's the integral (BTW, it's part of a double integral):

\int e^{x/y} dx

The answer is: ye^{x/y} dy but I don't understand why.

Wouldn't it be in the form e^u, with u being x/y, and du being 1/y dy? If so, then the answer should be e^{x/y}/y dy, right? That's wrong I guess, because the rest of the integral is pretty much impossible to do if that's the answer. So basically, my question is, why is the answer the answer? Thanks!

 

[Galileo]

You're integrating wrt x, so just consider y constant.
The derivative of yexp(x/y) wrt x is exp(x/y) by the chain rule, so that's the correct answer.

If you use substitution u=x/y, then you should replace dx with ydu (not use dy, since you're integrating wrt x).

 

[steelphantom]

You're integrating wrt x, so just consider y constant.
The derivative of yexp(x/y) wrt x is exp(x/y) by the chain rule, so that's the correct answer.

If you use substitution u=x/y, then you should replace dx with ydu (not use dy, since you're integrating wrt x).

Ah! That makes sense, especially since there's already a dy in the problem. No use having two of them. Thanks for the help!