# Prove two integral identities???

• Apr 23rd 2013, 04:31 AM
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!!
• Apr 23rd 2013, 04:40 AM
chiro
Re: Prove two integral identities???
Hey jarvisyang.

What substitution did you use for the first integral?