How do you integrate with respect to a function?

[Char. Limit]

Homework Statement

So I thought of an interesting problem, and here it is:

Solve \int \frac{1}{f'(x)} df(x) = g(x) for f(x).

Now, I checked Wolfram-Alpha to see if an answer existed, and they gave me this:

f(x) = c_1 e^{\int \frac{1}{g(\xi)} d\xi}

But, you know that Wolfram-Alpha doesn't show steps. So I want to know how they got from start to finish. I checked the solution, and it seems to work, but how did they get there?

Note that I have taken all three Calculus classes, but no differential equations experience.

EDIT: As a further question, how do you integrate with respect to a function? Is that even defined?

 

[Dick]

d(f(x)) is the differential of f(x). It's f'(x)*dx. That makes the left side pretty easy to integrate.

 

[I like Serena]

d(f(x)) is the differential of f(x). It's f'(x)*dx. That makes the left side pretty easy to integrate.

It also means that the solution contains any function f that is differentiable and has a non-zero first derivative, and any function g, as long as that function is of the form g(x)=c.

Wolfram's solution does look a bit weird though.
If you substitute g(x)=c, it works out to f(x)=c1.exp(ln(x))=c1.x.
It seems to me that Wolfram's answer is wrong.
It allows for any g, which is not true.
And it finds a specific solution for f, which is too restrictive.

Funny though =).