Is there anyone who can help me explaining the following integration by parts problem?
$$\iint \left(\frac{\partial f}{\partial t} \frac{\partial g^2}{\partial x^2} + \frac{\partial g}{\partial t} \frac{\partial f^2}{\partial x^2}\right)h(t,x) dt dx$$
where $f$, $g$ and $h$ are all functions in $\mathbb{R}_{>0}\times\mathbb{R}$ where $h$ is smooth with compact support in $\mathbb{R}_{>0}\times\mathbb{R}$.
I found the following expression but I am completely overburdened with these multiple variables...
$$\iint \left(\frac{\partial f}{\partial x} \frac{\partial g}{\partial x} \frac{\partial h}{\partial t} - \frac{\partial h}{\partial x} \frac{\partial f}{\partial x} \frac{\partial g}{\partial t} - \frac{\partial g}{\partial x} \frac{\partial h}{\partial x} \frac{\partial f}{\partial t}\right) dt dx$$
Thanks in advance!