1. The following integral identity holds

$\displaystyle \dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{ \sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)x}{a \sqrt{a^{2}-x^{2}}}+x\intop_{x}^{a}\dfrac{d\rho}{\sqrt{\rho^{2 }-x^{2}}}\dfrac{d}{d\rho}\left[\dfrac{F(\rho)}{\rho}\right]$

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

$\displaystyle \dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{ \sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)a}{x \sqrt{a^{2}-x^{2}}}+\dfrac{1}{x}\intop_{x}^{a}\dfrac{\rho d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}F(\rho)$

I can not figure out the second identity.Is there anybody can help me?I'm waiting for your excellent proof!!