This is from the complex analysis written by Gamelin, given a function $u(x,y)$ that is harmonic, follow Gamelin's argument, we found its harmonic conjugate $ v(x, y) = \int_{y_0}^{y} \frac{\partial u}{\partial x}(x,t)dt - \int_{x_0}^{x} \frac{\partial u}{\partial y}(s,y_0) + C$ how to verify that $v(x,y)$ is actually harmonic?
I know we can prove this by showing that $f = u+iv$ is analytic, but I can't prove that $\partial v / \partial x = -\partial u / \partial y$
By using the Leibniz rule for differentiating under the integral sign, and the FTC, we have \begin{align} \dfrac{\partial v}{\partial x}(x,y) &= \int_{y_0}^y \frac{\partial^2 u}{\partial x^2}(x,t) \, dt - \dfrac{\partial u}{\partial y}(x,y_0) \\ &= \int_{y_0}^y -\frac{\partial^2 u}{\partial y^2}(x,t) \, dt - \dfrac{\partial u}{\partial y}(x,y_0)\tag{$u$ is harmonic} \\ &= -\dfrac{\partial u}{\partial y}(x,y) \end{align} where in the last line I used the FTC (or it's corollary, or the second FTC whatever you wanna call it) again.