Integrating the product of an exponential and a first derivative

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Discussion Overview

The discussion revolves around finding an alternative method to integrate the product of an exponential function and the derivative of another function, specifically the integral of exp(x)*f'(x). Participants explore various techniques, including integration by parts, and express a desire for a simpler or more clever approach due to the complexity of the function f(x).

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about integration tricks beyond integration by parts for the integral of exp(x)*f'(x).
  • Another participant demonstrates integration by parts, resulting in exp(x)*f(x) - Int[exp(x)*f(x), dx], but expresses that this does not simplify the problem due to the complexity of f(x).
  • Some participants question the nature of f'(x) and suggest that knowing more about it might help in finding a solution.
  • There is a suggestion that there may not be an analytic way to integrate such a complicated function, proposing numerical methods or approximations instead.
  • One participant reiterates the hope for a clever trick related to the first derivative, while another doubts the existence of such a trick if the function is as complicated as its derivative.
  • Integration by parts is mentioned again as a potential method, but it is acknowledged that it leads back to the original complexity.

Areas of Agreement / Disagreement

Participants express a mix of agreement on the challenges posed by the complexity of f(x) and disagreement on whether a simpler method exists. The discussion remains unresolved regarding the existence of a clever integration technique.

Contextual Notes

Participants note the complexity of the function f(x) and its derivative, which may limit the effectiveness of standard integration techniques. There is also mention of the potential need for numerical methods or approximations, indicating limitations in finding an analytic solution.

bventer
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Hi, besides integration by parts, does anyone know of a simple integration trick to solve the integral (wrt x) of exp(x)*f'(x)?
 
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Could you show me how it's done by integration by parts?
 
Using integration by parts I get: exp(x)*f(x) - Int[exp(x)*f(x), dx]. But that doesn't really help me due to the complexity of the function f(x). I was hoping there might be a clever trick to exploit the fact that part of my integrand is a first derivative?
 
that doesn't really help me due to the complexity of the function f(x).
Why not showing us what is f '(x) ?
 
bventer said:
Using integration by parts I get: exp(x)*f(x) - Int[exp(x)*f(x), dx]. But that doesn't really help me due to the complexity of the function f(x). I was hoping there might be a clever trick to exploit the fact that part of my integrand is a first derivative?

Oh sorry, I missed the ' in f'(x) and thought you tried to find a solution to the integral of exf(x).

I can't think of another way to show it besides that technique, maybe someone else can.
 
To JJacquelin: I you wish, see attached jpg (apologies, but I am not well versed in Latex)
 

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Mentallic said:
Oh sorry, I missed the ' in f'(x) and thought you tried to find a solution to the integral of exf(x).

I can't think of another way to show it besides that technique, maybe someone else can.
Ok, thanks for having a look Mentallic
 
bventer said:
To JJacquelin: I you wish, see attached jpg (apologies, but I am not well versed in Latex)

How is the y(H) function defined?
 
Mentallic said:
How is the y(H) function defined?
y(H) is a quadratic: a0 + a1*H + a2*H^2
 
  • #10
Probably, there is no analytic way to integrate a so complicated function.
Better think to use numerical calculus, or approximations if it is a problem of physics.
 
  • #11
Thanks JJacquelin, but I did say that I was hoping there might be a clever trick to exploit the fact that part of my integrand is a first derivative...
 
  • #12
I was hoping there might be a clever trick to exploit the fact that part of my integrand is a first derivative...
I doubt that a "clever trick" exists. By the way, the first derivative of what function ?
If the function is as complicated as its derivative,then there is few hope.
 
  • #13
bventer said:
Thanks JJacquelin, but I did say that I was hoping there might be a clever trick to exploit the fact that part of my integrand is a first derivative...
Yes, there is! It is precisely the "integration by parts", letting dv= f'(x)dx that you initially did.
 
  • #14
Thanks, but unless I'm missing something this takes me back to the point I mentioned earlier in the thread: exp(x)*f(x) - Int[exp(x)*f(x), dx], which doesn't really help me due to the complexity of the function f(x).
 

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