How to Evaluate the Integral in the PDE Solution for U(x,t)?

[moo5003]

Homework Statement

U is a function of x and t

d/dt(U) = d/dx(U) + V(x,t)U
U(x,0) = f(x)

Suppose:
U(x,t) = e^(Integral from 0 to 1 [V(x+s,t-s)]ds) * f(x+t)

Show directly (no change of variables) that this solves the above PDE
Show using change of variables that this solves the above PDE letting
Alpha = x+t
Gamma = x-t

The Attempt at a Solution

My main question is how to evlauate d/dt { Integral from 0 to 1 [V(x+s,t-s)]ds }

 

[D H]

Use the Liebniz integral rule, described in these articles at http://en.wikipedia.org/wiki/Leibniz_integral_rule" .