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I'm just trying to understanding something here. I was taught integration by parts by my professor in what looks to be like an untraditional way.

From my understanding, the theorem states:

∫udv = uv - ∫vdu

We were given an example in class of:

∫e

^{x}sin(x)dx

=∫e

^{x}∫sin(x)dx - ∫[(e

^{x})'∫sin(x)dx]dx

=-e

^{x}cos(x) + ∫e

^{x}cos(x)dx

I don't understand how this is the same thing as if I were to use, e

^{x}as u and sin(x) as v, since the outcome would then be:

∫e

^{x}sin(x)dx

=e

^{x}sin(x)dx - ∫e

^{x}sin(x)dx

Any help understanding this professors method would be great, thanks!