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Calculate the following symbolic integral?

Nick Arnold
Posted 5 years ago

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?
POSTED BY: Nick Arnold
10 Replies
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Did you have in mind something like this?

D[InverseFunction[Composition[Log, g]][x], x] /. x -> Log[g[y]]
POSTED BY: Gianluca Gorni

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 doesnÂ’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.
POSTED BY: John Doty

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 $g(x)$ ?

POSTED BY: Henrik Schachner
Nick Arnold
Nick Arnold
Posted 5 years ago

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)]
POSTED BY: Nick Arnold

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

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

.
POSTED BY: Marvin Ray Burns

This variant gives a useful result:

Integrate[g'[x]/g[x], x]
POSTED BY: Gianluca Gorni

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]]

$$\int \frac{g (x)}{g'(x)} \, dx.$$
POSTED BY: Marvin Ray Burns
Nick Arnold
Nick Arnold
Posted 5 years ago

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

POSTED BY: Nick Arnold

What is g[x]?
POSTED BY: Murray Eisenberg
Nick Arnold
Nick Arnold
Posted 5 years ago

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

POSTED BY: Nick Arnold
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