How Does the General Leibniz Formula Apply to Integrals with Variable Limits?

[kid_abraxis]

can anyone shed some light on this little monster?

 

[Defennder]

What you should do is to use the chain rule. Denote the upper limit of the integral (the limit which itself is an integral) as y(x). Then you should be able to see that dF/dx = dF/dy dy/dx.

 

[kid_abraxis]

i'll take a look. ta

 

[HallsofIvy]

The general Leibniz' formula is
$$\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} F(x,t)dt= \frac{d\beta(x)}{dx}F(x,\beta(x))- \frac{d\alpha(x)}{dx}F(x,\alpha(x))+ \int_{\alpha(x)}^{\beta(x)}\frac{\partial F(x,t)}{\partial x} dt$$