I came across the following identity, without any explanation or proof:
$$\int_{t_0}^{t_*}dt\int_{t}^Tdx\frac{\partial f(t,x)}{\partial t}=\int_{t_0}^{t_*}dtf(t,t)+\int_{t_*}^Tdxf(t,x)-\int_{t_0}^Tdxf(t,x).$$
I have been trying to find a "proof" (does not need to be entirely rigorous) for quite some time now but can't seem to find an explanation. The function $f$ is a well-behaving continuous function.
Any tips/help is welcome!
With the help of @AnneBauval 's comment and drawing a graphical representation of the integration region, I was able to come up with the following derivation:
$$\int_{t_0}^{t_*}dt\int_t^Tdx\frac{\partial f(t,x)}{t}=\int_{t_0}^Tdx\int_{t_0}^xdt\frac{\partial f(t,x)}{t}-\int_{t_*}^Tdx\int_{t_*}^xdt\frac{\partial f(t,x)}{t}$$ By the fundamental theorem of calculus $$=\int_{t_0}^Tdx[f(x,x)-f(t_0,x)]- \int_{t_*}^Tdx[f(x,x)-f(t_*,x)]$$ Collecting $f(x,x)$ yields $$=\int_{t_0}^{t_*}dxf(x,x)-\int_{t_0}^Tdxf(t_0,x)+\int_{t_*}^Tdx f(t_*,x)$$
Such as was mentioned in the comments, the identity contains a typo.