Another Integration Formula

  • #1
gleem
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According to the author of this article, this integration formula is well-known but rarely taught. Do you know it?



where x0 is a constant and f-1(x) is the inverse function of f(x).
 
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  • #2
Is there a typo somewhere? I get for

Where am I wrong?
 
  • #3
gleem said:
According to the author of this article, this integration formula is well-known but rarely taught. Do you know it?



where x0 is a constant and f-1(x) is the inverse function of f(x).
It's a combination of parts and substitution. You can start from the integral of an inverse function. With , do a change of variables, followed by parts.
 
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  • #4
fresh_42 said:
Is there a typo somewhere? I get for

Where am I wrong?
The integral of is .
 
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  • #5
PeroK said:
It's a combination of parts and substitution. You can start from the integral of an inverse function. With , do a change of variables, followed by parts.
You can also just take the derivative of the RHS, and verify that it is f(x). I note that I have never seen this before. It looks cute, but I wouldn't know how useful it is without playing with some examples myself.

[The derivative comes out ]
 
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  • #6
PAllen said:
It looks cute, but I wouldn't know how useful it is without playing with some examples myself.
This looks like one of those integration identities that might find use on some ubsurd integral challenges (which I enjoy), but probably not much use anywhere else. I would be interested to see if it does show up in someones research somewhere.
 
  • #7
Mondayman said:
This looks like one of those integration identities that might find use on some ubsurd integral challenges (which I enjoy), but probably not much use anywhere else. I would be interested to see if it does show up in someones research somewhere.
It could be useful, but it doesn't have a lot of independent value. It's just the full substitution, followed by parts. In that sense, it's nothing new.
 

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