Finding ∂/∂x ∫₀ʸ u dy using Leibniz rule and integration by parts

[Dkie]

Homework Statement

Using Leibniz rule and integration by parts, solve \frac{\partial}{\partial x} \int_0^y u dy.

Homework Equations

u = U(x) f' (\eta)

\eta = \eta(x,y) = y g(x)

 

[LCKurtz]

Homework Statement

Using Leibniz rule and integration by parts, solve
$$\frac{\partial}{\partial x} \int_0^y u dy.$$

You need to enclose your tex in tex tags (without the space) [ tex]...[/tex].

Is perchance that dy supposed to be dx? And if so, what have you tried?

 

[HallsofIvy]

In particular is u a function of both x and y?

I don't see any need for "integration by parts". Leibniz' rule says
$$\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x,y)dy= \frac{d\beta}{dx}f(x,\beta(x))- \frac{d\alpha}{dx}f(x,\alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f}{\partial x}dy$$

So whether the derivative is with respect to x or y, that should be enough.
(Unless u is some special function you didn't mention.)