A bug in the book?

Hi Experts,
This is very strange:
The book I am reading is doing some simple math with mistake?
The load p(x) is applied at the surfaces of a crack (from 0 to a), the
released enagy
U = 0.5int[p(x)v(x,a)]dx (from 0 to a), fine.
However, the book then differentiate the equation with respect to a
and got
dU/da = 0.5int[p(x)dv(x,a)/da]dx (from 0 to a).
Is this forgot 'a' is a variable now in the integration! Why?
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You forgot to define v(x,a) so it is pretty frustrating to try to reply. What do you mean by a "forgot 'a'" in your sentence?
What is the value of the function v(x=a,a) or v(a,a)?
v(x,a) is probably the Green's function for displacement (v) at point x due to a load at point x for a crack of half length, a. However the whole thing appears garbled.
Can you explain the physical significance of the terms of the equation?
If the author didn't explain the physical significance of the equation terms, then this sounds like time to find a different text that may explain it better.
You have the book in front of you, probably none of us does.
Look in a good calculus book for how to differentiate a finite integral, where one of the limits is a function of the variable of differentiation.
I'm gonna quit here.
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