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The process of differentiating a known standard function
can be achieved by parsing the function definition, and applying
the appropriate rules of differentiation to each step in its
definition. There are no specific anti-differentiation rules;
we have to use the differentiation rules backwards. And these
allow us to anti-differentiate large classes of standard functions,
but not all of them. We can recognize some functions almost immediately as derivatives
of others. For example, we can anti-differentiate any polynomial
in x by applying the rule to differentiate powers backwards.
Thus an anti-derivative of It turns out, that there are more tricks which allow us to
differentiate any rational function On the other hand there are simple looking standard functions
that we cannot anti-differentiate, because their anti-derivatives
do not happen to be standard functions. There used to be a difference of opinion among teachers of
calculus as to whether students should be made to learn all
the tricks of anti-differentiation so that they could anti-differentiate
anything that could be anti-differentiated. Some felt that
the mark of mastery of calculus was to be able to anti-differentiate
strange looking functions by attempting to apply the various
tricks. Others pointed out that there are "integral tables"
which list essentially all anti-differentiable functions and
their anti-derivatives, so that this skill, whose development
is a wonderful exercise of ingenuity and memory, has very
limited practical value. At that time there were obstacles
that students faced in getting a copy of a reasonably complete
integral table, and in looking things up in huge ones, that
gave the edge by a small margin to the first viewpoint. Today,
however, there are commercially available programs, Maple,
MATLAB, and Mathematica (and perhaps others), which give you
detailed formulae for the anti-derivative of any function
you care to enter, and do so instantly; which seems to tilt
the argument toward the second viewpoint. Before doing so we should notice the important fact that the action of finding an anti-derivative is a linear operation (as differentiating is) so that an anti-derivative of a sum is the sum of anti-derivatives of its summands. |
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