When speaking in conversation, or writing up a paper, and applying the said technique on a function, is it correct to say "partially integrate"?
"We partially integrate X..."
"We integrate X by parts..."
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I would say "We integrate X by parts."
If someone said "partial integration" I may be inclined to try to think of some sort of inverse of partial differentiation.
On the other hand, one does see "partial integration" for "integration by parts" sometimes; I am guessing this is literally translated from some other language.