Show that if f(x) is continuous and u(x) and v(x) are differentiable functions, then:
d/dx ∫_v(x)^u(x) f(t)dt=u'(x)f(u(x))-v'(x)f(v(x))
It's just the chain rule. If
F(t) = ∫f(t) dt
then
∫[v(x),u(x)] f(t) dt
= F(u)-F(v)
so, taking derivatives, that gives you
f(u) u' - f(v) v'