its from thermodynamics...but i dont think you really need to understand thermodynamics to figure out what math trick they used to get from the first integral to the second integral

i have been looking at this equation for hours and cannotfigure out how that partial differential and the 'dT' just disappeared!!
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 Hey racnna and welcome to the forums. Are you familiar with identities involving the fundamental theorem of calculus?
 hey chiro...no im not..or maybe i have just forgotten....i just googled but cant seem to find any useful info....can you please explain this identity or link me to a place that explains it? thanks!

they evaluated the dt integral using the fact that $\int_a^b \partial _x f(x,y) dx = f(b,y)-f(a,y)$