How Can These Two Integral Identities Be Proven?

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Discussion Overview

The discussion revolves around proving two integral identities involving a function \( F(\rho) \) and its derivatives. The identities are presented in the context of calculus, specifically focusing on differentiation under the integral sign and integration by parts. Participants are seeking assistance in proving the second identity while one claims to have already proven the first.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents two integral identities and requests help with proving the second identity, suggesting that integration by parts may be useful for the first identity.
  • Another participant emphasizes the forum's policy that assistance requires demonstrable effort from the person seeking help.
  • A participant insists they have made significant efforts to prove the first identity but struggles with the second, indicating ongoing contemplation without resolution.
  • Further encouragement is provided to show attempts at solving the problem, regardless of their perceived quality, to facilitate better assistance from others.
  • A participant acknowledges the advice received and expresses intent to share their attempts in future posts.

Areas of Agreement / Disagreement

There is no consensus on the proof of the second identity, as participants express differing views on the necessity of demonstrating effort before receiving help. The discussion remains unresolved regarding the specific proof of the second integral identity.

Contextual Notes

Participants have not provided specific attempts or methods for proving the second identity, and there is a lack of detailed exploration of the mathematical steps involved in either identity.

jarvisyang
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Prove two integral identities?

1. The following integral identity holds
[tex]\dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{\sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)x}{a\sqrt{a^{2}-x^{2}}}+x\intop_{x}^{a}\dfrac{d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}\left[\dfrac{F(\rho)}{\rho}\right][/tex]
Hints: this can easily proved by applying ingtegration by parts to the right hand side of the identity
2. But the following can also hold
[tex]\dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{\sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)a}{x\sqrt{a^{2}-x^{2}}}+\dfrac{1}{x}\intop_{x}^{a}\dfrac{\rho d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}F(\rho)[/tex]
I can not figure out the second identity.Is there anybody can help me?I'm waiting for your excellent proof!
 
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We are not allowed to help in problems unless you demonstrate an effort to solve the problem yourself.
 
pwsnafu said:
We are not allowed to help in problems unless you demonstrate an effort to solve the problem yourself.
Actually, I have made a lot efforts. The first identity has been proved by myself. But as for the second identities, I have been thinking for a long time and I still can not figure it out.
 
jarvisyang said:
Actually, I have made a lot efforts. The first identity has been proved by myself. But as for the second identities, I have been thinking for a long time and I still can not figure it out.

Well, you know what you do in that case don't you? Show us what you tried even if it's stupid-looking. You know good cooks try again don't you? Yeah, they mess up but they don't get discouraged, then try the recipie again, and eventually they cookin' with kerosene and you wonder how they got so good. Try the recepie even if you burn the dish. The trying part is important.
 
jackmell said:
Well, you know what you do in that case don't you? Show us what you tried even if it's stupid-looking. You know good cooks try again don't you? Yeah, they mess up but they don't get discouraged, then try the recipie again, and eventually they cookin' with kerosene and you wonder how they got so good. Try the recepie even if you burn the dish. The trying part is important.

OK.Thank you for your suggestion, jackmell. I've got it. I will post my question together with my efforts or scripts next time.
 

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