Harmonic holomorphic function in Ω

Question

$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$

$$v(x,y) = \int_0^1 (yu{\Tiny x} (tx,ty)-xu{\Tiny y} (tx,ty)) dt$$

Prove that there exists a holomorphic function $u+iv$ in $Ω$

I have the solution - can someone explain me the equalities to where i put the orange question marks? I dont understand why the first part of the term disappears.

Photo/solution

Answer 1

Let $g(t):= tu_y(tx,ty)$, then

$ \int_0^1 \frac{d}{dt}(tu_y(tx,ty)) dt = \int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$