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Since the integration of a function is the opposite of a the derivative of a function, and there are clear steps to follow when we want to find the derivative of a function, I thought there would be clear steps to follow too when we want to find the integral of a function.

What is it about finding the integration of a function that makes it impossible to come up with a guaranteed method to find it?

Edit: I am only considering elementary functions.

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marked as duplicate by Rahul, leonbloy, mauna, mrf, azimut Dec 9 '13 at 13:26

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Integration is the opposite of differentiation. The opposite of there being clear steps to follow is no clear steps to follow. shrugs – David H Dec 9 '13 at 12:25
There are clear steps to follow when we want to find the derivative of a function, only for a narrow class of functions. In general there are no clear steps to follow. The same holds for integrals. – Student Dec 9 '13 at 12:26
@David, what? I hope you are not serious. – Student Dec 9 '13 at 12:27
@Ivan I was being cute. I suppose I could have made that more transparent, but it's late and I'm tired. =p – David H Dec 9 '13 at 12:30
See also Why is differentiating mechanics and integration art? on MathOverflow. – Rahul Dec 9 '13 at 12:37
up vote 4 down vote accepted

Because the class of function you consider easy to differentiate—namely, the elementary functions—is not mapped onto itself. You are accustomed to thinking of elementary functions as being exactly those on which you "do calculus" because they are easy to differentiate, but a randomly chosen one may not be in the image of the derivative function, and therefore, not easy to integrate. The standard example $f(x) = e^{x^2}$ applies here; no simple example of a hard-to-differentiate function exists, by definition.

Note that without the assurance that functions are given by explicit formulas of a convenient form, neither integration nor differentiation is particularly easy.

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For special types of functions there are pretty clear cut methods for integrating (e.g. polynomials). Unfortunately, there is no general algorithm for creating integrals as there are functions that have no antiderivative. The classic example being $\ f(x)=e^{x^2}$.

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$e^{x^2}$ has no elementary derivative. It has obviously an antiderivative, being a continuous function... – Najib Idrissi Dec 9 '13 at 13:28

There is an algorithm for computing indefinite integrals in terms of elementary functions, when such representations exist: Risch's algorithm . It is a very complex algorithm which is now included in the major symbolic mathematics programs.

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There are clear steps to follow when we want to compute the derivative of a function, only for a narrow class of functions. In general no algorithm is known for differentiation. The same holds for integrals.

EDIT: If in the search for further information, turn to the link, kindly suggested by Chris Janjigian.

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It would be nice to add an example of such a "no clear steps to find the derivative" function. – Xoff Dec 9 '13 at 12:37
That would be really hard, as all elementary functions have a well known derivative, @Xoff. – Student Dec 9 '13 at 12:46
If you say "in general", then it must be easy to find an example. If not, then don't say "in general". – Xoff Dec 9 '13 at 12:56
So your answer to "why isn't there a fixed procedure" is "because there is no fixed procedure"? – Najib Idrissi Dec 9 '13 at 12:58
@JonathanY. I understand your point of view, because you think about the set of functions, and in general means "for almost all". But for the OP, I'm pretty sure it means "for almost all functions I will encounter", and it's not the same thing. If you can't give an example of something and you claim that this thing is almost always occurring, it's a real pedagogical problem. – Xoff Dec 9 '13 at 13:37

In order to answer your question, one has to clarify it. You presumably mean integrating explicitly elementary functions in terms of further elementary functions. Of course, a complete answer would required a precise definition of this term but we can regard it informally as meaning one which is obtained from the usual suspects (powers, trigonometric, exponential, logarithm, etc.) by simple operations (arithmetic, composition, etc.). The difference is then that while there are simple rules which describe what happens when differentiating, there are no such ones for integration. For example, integration by parts merely shifts the problem to a new integral---this might work in a particular case but not in general. You can see this effect when you use Mathematica which cannot integrate all elementary functions explicitly. By the way there is, rather surprisingly, a general theory which covers this topic and this is used in the algorithm employed by Mathematica.

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I see that the algorithm I mentioned is pinpointed in the answer below of Ihf which was posted while I was at the coal face. – mathuser5891 Dec 9 '13 at 12:51

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