Difference expressed as integral of differential?

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Difference expressed as integral of differential??

Hi all, I came across an equation in this form while trying to understand a paper:

[tex]f(t+T) - f(t) = \int_t^\(t+T\ [/tex][tex]\frac{d}{dt} f(t') \, dt'[/tex]

but I was unable to see how it can be true. If I bring the term [tex]\frac{d}{dt}[/tex] outside of the definite integral, it seems to work, but I don't think that is allowed? Can anybody help? Thanks!
 

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[tex]f(t+T) - f(t) = \int_t^\(t+T\ [/tex][tex]\frac{d}{dt} f(t') \, dt'[/tex]
I think it should be written with [tex]\frac{d}{dt'}[/tex] inside, not [tex]\frac{d}{dt}[/tex] .
 

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