Given that $\frac{\partial g_1}{\partial x_2} = \frac{\partial g_2}{\partial x_1}$ How can I simplify the following? $\int_0^{x_2}\frac{\partial g_2(x_1,y)}{\partial x_1}\,\mathrm dy.$
Note: If I just rewrite it as $\int_0^{x_2}\frac{\partial g_1(x_1,y)}{\partial x_2} \,\mathrm dy.$ Doesn't it become zero? Because seems like $\frac{\partial g_1(x_1,y)}{\partial x_2}$ should be zero, as $g_1$ is not a function of $x_2$.
If it is not zero, then can I write $\int_0^{x_2}\frac{\partial g_1(x_1,y)}{\partial x_2} \,\mathrm dy = \frac{\partial}{\partial x_2}\int_0^{x_2} g_1(x_1,y) \,\mathrm dy$ ?