Calculate the following symbolic integral?

Questions
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1

Consider the following code:

f[x_] := g[x]/g'[x] 

FullSimplify[Integrate[f[x], x]]

So I have the code above but when I run it it gives me back some random expression. I’m trying to make it return something in terms of g(x) but it won’t.

How would I make it give me F(x) in terms of g(x)?

1/4 (x^2 + 2 Log[x])

^ What it returns when I run the code. Where is it getting that from?

2

What is g[x]?

3

I don’t have it set to anything, I want it to be generic and just give F(x) in terms of g(x) and it’s derivatives so that that stuff can just be plugged in. I want it to just be symbolic

4

There exists no formula for an integral that works for all g(x), So, if you quit your kernel, you’ll see that Mathematica simply returns an unevaluated form:

Enter

 Quit[];

.

Then enter

 f[x_] := g[x]/g'[x]
FullSimplify[Integrate[f[x], x]]
\int \frac{g (x)}{g'(x)} \, dx.
5

This variant gives a useful result:

Integrate[g'[x]/g[x], x]
6

Ah thanks, quitting the kernal worked for that. Sucks that there’s no general solution for that form of integral

7

Sucks that there’s no general solution for that form of integral

There are even simpler forms that “do not work”; what kind of output in terms of g(x) you would expect from

\int g(x) \mbox{d}x

without specifying g(x) ?

8

Fair enough, I was trying to find a general form for the function whose derivative is the inverse of the derivative of log[g(x)]

9

Nick Arnold:
Try Gianluca Gorni’s input. I must have forgot that there was a standard solution for

Integrate[g'[x]/g[x], x]

.

10

A bit of impetuous formal manipulation perhaps illuminates. In the integrand (1/g) (dg/dx) dx, the dx cancel, leaving dg/g. The dependence on x has disappeared. This doesn’t happen with the integrand you want to integrate, so the integral cannot be performed without knowledge of the functional relationship between g and x.

11

Did you have in mind something like this?

D[InverseFunction[Composition[Log, g]][x], x] /. x -> Log[g[y]]