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If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product of integrals can't be expressed in general in terms of the integral of the products, and forget about composition! Why is this?

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@Patrick: This doesn't exactly answer your question but is on related lines. math.arizona.edu/~mleslie/files/integrationtalk.pdf –  user17762 Feb 5 '11 at 20:42
This question reminded me of a tangentially related post on MathOverflow: "... on formulas differentiation is nice and integration is hard, but on computable functions differentiation is hard and integration is nice." -- Jacques Carette –  Rahul Feb 5 '11 at 21:34
@sivaram: that's an interesting slideshow! The last slide is hilarious. –  Myself Feb 6 '11 at 2:15
I think most analysts out there would say integration is much easier than differentiation...but of course they have a different thing in mind than your question. –  Matt Feb 6 '11 at 2:39
To rephrase Jacques's quote: differentiation is symbolically easy but numerically hard, while integration is numerically easy but symbolically hard. –  J. M. Apr 20 '11 at 11:03
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5 Answers

up vote 38 down vote accepted

Here is an extremely generic answer. Differentiation is a "local" operation: to compute the derivative of a function at a point you only have to know how it behaves in a neighborhood of that point. But integration is a "global" operation: to compute the definite integral of a function in an interval you have to know how it behaves on the entire interval (and to compute the indefinite integral you have to know how it behaves on all intervals). That is a lot of information to summarize. Generally, local things are generally much easier than global things.

On the other hand, if you can do the global things, they tend to be useful because of how much information goes into them. That's why theorems like the fundamental theorem of calculus, the full form of Stokes' theorem, and the main theorems of complex analysis are so powerful: they let us calculate global things in terms of slightly less global things.

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Patrick asked about the integration of of "function terms", i.e. finite expressions. In your answer you omitted to say that such an expression is a global object to begin with. –  Christian Blatter Feb 6 '11 at 13:26
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The family of functions you generally consider (e.g., elementary functions) is closed under differentiation, that is, the derivative of such function is still in the family. However, the family is not in general closed under integration. For instance, even the family of rational functions is not closed under integration because you $\int 1/x = \log$.

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It seems to be hard to find families of functions that are closed under integration. –  lhf Oct 12 '13 at 2:55
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Answering an old question just because I saw it on the main page. From Roger Penrose (Road To Reality):

... there is a striking contrast between the operations of differentiation and integration, in this calculus, with regard to which is the ‘easy’ one and which is the ‘difficult’ one. When it is a matter of applying the operations to explicit formulae involving known functions, it is differentiation which is ‘easy’ and integration ‘difficult’, and in many cases the latter may not be possible to carry out at all in an explicit way. On the other hand, when functions are not given in terms of formulae, but are provided in the form of tabulated lists of numerical data, then it is integration which is ‘easy’ and differentiation ‘difficult’, and the latter may not, strictly speaking, be possible at all in the ordinary way. Numerical techniques are generally concerned with approximations, but there is also a close analogue of this aspect of things in the exact theory, and again it is integration which can be performed in circumstances where differentiation cannot.

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Just for the record: Penrose discusses these matters on p. 103-120. Yet he gives no good reason why this is so (esp. for the symbolic case). –  vonjd May 31 '11 at 9:22
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In the MIT lecture 6.001 "Structure and Interpretation of Computer Programs" by Susskind and Abelson this contrast is briefly discussed in the terms of pattern matching. See the lecture video (at 3:56) or alternatively the transcript (p. 2). The relevant part is quoted below. (The book does not seem to provide further explanation.)

Edit: Apparently they discuss the Risch algorithm. I recommend reading some answers to the same question on mathoverflow.SE: Why is differentiating mechanics and integration art?

And you know from calculus that it's easy to produce derivatives of arbitrary expressions. You also know from your elementary calculus that it's hard to produce integrals. Yet integrals and derivatives are opposites of each other. They're inverse operations. And they have the same rules. What is special about these rules that makes it possible for one to produce derivatives easily and integrals why it's so hard? Let's think about that very simply.

Look at these rules. Every one of these rules, when used in the direction for taking derivatives, which is in the direction of this arrow, the left side is matched against your expression, and the right side is the thing which is the derivative of that expression. The arrow is going that way. In each of these rules, the expressions on the right - hand side of the rule that are contained within derivatives are subexpressions, are proper subexpressions, of the expression on the left - hand side.

So here we see the derivative of the sum, with is the expression on the left - hand side is the sum of the derivatives of the pieces. So the rule of moving to the right are reduction rules. The problem becomes easier. I turn a big complicated problem it's lots of smaller problems and then combine the results, a perfect place for recursion to work.

If I'm going in the other direction like this, if I'm trying to produce integrals, well there are several problems you see here. First of all, if I try to integrate an expression like a sum, more than one rule matches. Here's one that matches. Here's one that matches. I don't know which one to take. And they may be different. I may get to explore different things. Also, the expressions become larger in that direction. And when the expressions become larger, then there's no guarantee that any particular path I choose will terminate, because we will only terminate by accidental cancellation. So that's why integrals are complicated searches and hard to do.

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Think about what the two represent. If you have some function (it need not be too complicated) such as $f(x) = x^2 \sin(x)$, what does $\frac{df}{dx}(3)$ and $\int_0^3 f(s) ds$ mean?

Suppose you don't know any theory, and you are given the graph of the function, which will be easier to find?

Then it's not surprising that you can calculate the derivative of complicated functions whereas the integral might not even have a closed form.

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