So I can’t quite decide what to call integration/antidifferentiation in the notes. I have always used integrate with the understanding that it means to compute the Riemann integral and thereby compute the antiderivative by FTC. That is, writing the indefinite integral as a shortcut for the actual integral:

But it seems that when students hear “integrate” they really think just of symbolic antidifferentiation. I could change all instances of “integrate” to “antidifferentiate.” But in places it just sounds bad. I feel like I want to actually integrate at places because I want to “compute” the actual functions. To me “antidifferentiation” smells of trying to get a closed form expression in terms of elementary functions or some such. As an analyst I want to always just “integrate.”

Also if I change to the word “antidifferentiate,” then the whole “integrating factor method” sounds off.

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The following example made me aware that when calculating with indefinite integrals, the equalities really aren’t equalities but rather equivalence relations: we write f = g if f’ = g’.

∫ 1/x dx

= ∫ 1 * 1/x dx = { partial integration }

= x * 1/x – ∫ x * (-1/x²) dx

= 1 + ∫ 1/x dx

Comment by Per Persson — August 6, 2010 @ 6:59 pm |