# Prove two integral identities?

1. Apr 23, 2013

### jarvisyang

Prove two integral identities???

1. The following integral identity holds
$$\dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{\sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)x}{a\sqrt{a^{2}-x^{2}}}+x\intop_{x}^{a}\dfrac{d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}\left[\dfrac{F(\rho)}{\rho}\right]$$
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
$$\dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{\sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)a}{x\sqrt{a^{2}-x^{2}}}+\dfrac{1}{x}\intop_{x}^{a}\dfrac{\rho d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}F(\rho)$$
I can not figure out the second identity.Is there anybody can help me?I'm waiting for your excellent proof!!

2. Apr 23, 2013

### pwsnafu

We are not allowed to help in problems unless you demonstrate an effort to solve the problem yourself.

3. Apr 23, 2013

### jarvisyang

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.

4. Apr 24, 2013

### jackmell

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.

5. Apr 27, 2013

### jarvisyang

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