Leibniz rule with multivariable function

Question

We have:

$$F(u,v)=\int_0^v\frac{1-cos(ut)}{t} dt $$

Find $$F_u$$ and $$F_v$$

EDIT (solution):

$$ F_u=\int_0^v \frac{\partial}{\partial u} \frac{-cos(ut)}{t}dt = \int_0^v sin(ut) dt = \frac{-cos(uv)}{u}-\frac{(-cos(0)}{u}=\frac{1-cos(uv)}{u}$$

$$ F_v = \frac{1-cos(uv)}{v} $$ comes from the second theorem of calculus

Answer 1

We have

$F_v(u,v)= \frac{1-cos(uv)}{v}$ (why ??)

and

$F_u(u,v)=\int_0^v \frac{\partial}{\partial u}(\frac{1-cos(ut)}{t}) dt$

Now show that $F_u(u,v)= \frac{1-cos(uv)}{u}$