Integrals: A Question on Validity and Substitution

[ManyNames]

So, this is really a question. Suppose you have:

$$a(t)=\int_{\Omega} |\psi|^2$$

and then you have,

$$\frac{\partial t}{\int_{t_0}^{t_1} dt a(t)}$$

Surely this can't be right, because that would imply by substitution

$$\frac{\partial t}{\int_{t_0}^{t_1} dt \int_{\Omega} |\psi|^2}$$

Is that really allowed?

 

[ManyNames]

Can no one answer this for me?

 

[chrisk]

What is the differential of the integral in the first equation?