Why are there so many integrals that are impossible to calculate?

[wakwakwak3]

I understand that you can find the derivative of almost anything but I am perplexed by how many functions you cannot integrate. Does anyone here know a good set of criteria to follow in determining if a function has an anti derivative? What are all the classes of functions that are impossible to integrate? Finally, why is it that this occurs in math?

 

[mmm4444bot]

These are good research questions because you can easily find information by searching. (I doubt you'll find a complete list of functions that lack an elementary antiderivative, but there's still good information out there.)

Google search phrases like list of functions without antiderivatives, to start.

http://web.archive.org/web/20160612...tions-You-Cannot-Integrate-No-Antiderivatives

https://math.stackexchange.com/ques...t-a-function-has-no-elementary-antiderivative

https://www.quora.com/Is-there-any-...essed-in-terms-of-elementary-functions-or-not

https://www.quora.com/Why-can-we-differentiate-any-function-but-cant-integrate-any-function

Of course, if you see something specific that you don't understand, feel free to ask about it. :cool:

 

[pka]

I understand that you can find the derivative of almost anything but I am perplexed by how many functions you cannot integrate. Does anyone here know a good set of criteria to follow in determining if a function has an anti derivative? What are all the classes of functions that are impossible to integrate? Finally, why is it that this occurs in math?

You my dear fellow are in for quite a surprise. The domain of real valued functions is a wonderful and strange place. There are everywhere continuous functions that have derivatives nowhere. There are derivatives of other functions that have no integrals themselves. Now the good news is that these functions are rarely ever encountered in basic calculus courses. It maybe that in the first two courses in calculus one may not be exposed to advanced techniques in integration such as using infinite series to express a function. But for some of the more difficult function there are advanced methods. So please learn the basics. Wait for the good material.