Is there an Integration Rule to Evaluate F(F'(x))

[Al-Layth]

I know there is no general rule to evalute F(G(x)) but what if the inner function is a derivative of the outer. ?

[math]\int{F(F'(x))}dx=?[/math]

 

[Cubist]

I know there is no general rule to evalute F(G(x)) but what if the inner function is a derivative of the outer. ?

[math]\int{F(F'(x))}dx=?[/math]

There's no general rule to integrate F( F'(x) ) with respect to x

There is a way to integrate an expression of the form Z(x) * F( F'(x) ) where Z(x) can be written in terms of F. If you're interested, then you can easily explore this for yourself. Start by differentiating H( G(x) ). You want the result of this to look like Z(x) * F( F'(x) ). Using this information you can then determine a H and G in terms of F. This will then tell you Z in terms if F, and it will also give you a general rule.

The above is an interesting thing to consider, but I think that learning to spot a good change of variable (that makes the integration possible) would be a more standard approach.

 

[Al-Layth]

There's no general rule to integrate F( F'(x) ) with respect to x

There is a way to integrate an expression of the form Z(x) * F( F'(x) ) where Z(x) can be written in terms of F. If you're interested, then you can easily explore this for yourself. Start by differentiating H( G(x) ). You want the result of this to look like Z(x) * F( F'(x) ). Using this information you can then determine a H and G in terms of F. This will then tell you Z in terms if F, and it will also give you a general rule.

The above is an interesting thing to consider, but I think that learning to spot a good change of variable (that makes the integration possible) would be a more standard approach.

how are you certain there is no general rule for it

Also I will try what you said thanks for that. more general rules I have the better

 

[Deleted member 4993]

how are you certain there is no general rule for it

We are never certain about anything. Depth of our knowledge changes and sphere of certainty changes.

People were certain - Earth is flat. Not anymore....

People were certain - Sun revolves around the earth. Not anymore....

We should probably add a part - As far as we know.....

However typing that before every proclamation will use up too many electrons. We learn by repeatation (practice) and experimentation (discovery).

 

[topsquark]

how are you certain there is no general rule for it

Also I will try what you said thanks for that. more general rules I have the better

If you put in enough effort you can probably write a formula for just about anything. There might indeed be a general rule for it, but I suspect such a rule would have an infinite number of terms so it wouldn't be of all that much use, in general.

For example, the most reasonable way to start would be to make the substitution u = F'(x). So du = F''(x) dx. In order to put this into the integrand we need to solve u = F'(x) for x and put that into du = F''(x) dx. Already we are running into some major problems. But they could potentially be solved by an infinite series approach. But that means that just at the first step we are using an infinite series. And the next step will give the same headaches. So another infinite series, etc. But I don't see why it couldn't be done in principle. It's just going to be so complicated that I doubt anyone could use it unless things work out just right. Such as for something simple like [imath]F(x) = x^2[/imath].

-Dan

 

[Cubist]

how are you certain there is no general rule for it

Well, there is a rule/ method, but I wouldn't call it general It's just the quite specific case when Z(x)=1 in my post#2

[math]\int Z(x) \times F\left( F'(x) \right) dx[/math]
You'll see that this is already fairly specific, IMO anyway, before constraining Z(x) to be 1

Also I will try what you said thanks for that.

Please do. This will reveal Z(x). Follow the advice in post#2. Show us your work (or tell us where you get stuck). It should be quite quick. And, doing this will also lead to an answer to your other thread integration of the product f(x)f(x)

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EDIT: There might actually be a general way to perform the integral (as per posts#2 and #3) after having thought about it (within the 30min edit time ). My post#2 method probably just explores situations where the integral has a particular type of result.