Is there any way to evaluate the integral: $$ \int_0^{1}e^{-x^2}\,\mathrm{d}x $$ using only basic integration techniques (Basic formulas, integration by parts, Substitution and change of variables)
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1not at all, only if the upper limit would be $\infty$ But this integral is very well known and has its special name. It is coined the Errorfunction. $\sqrt{\pi}/2 erf[1]$ is the best you can do – tired Feb 10 '15 at 13:47
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@tired And even then "only" with double integrals (at least as far as I know), otherwise gamma function, complex integration and stuff can be used. – Timbuc Feb 10 '15 at 13:48
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but is there any proof that this integral cannot be solved using only basic techniques ? – Mehdi Zibout Feb 10 '15 at 13:49
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@MehdiZibout http://en.wikipedia.org/wiki/Risch_algorithm Note that the algorithm is over 100 pages long, which should say something about how complicated this question really is. Also, this is just for indefinite integrals; the problem of detecting definite integrals that can be solved by some "trick" (like the infinite Gaussian integral) is even harder. – Ian Feb 10 '15 at 14:02
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As you can see from the link, such things can be proved, but to show the details of such a proof would not give you any understanding (unless you are the kind of person who can gain understanding by scanning a computer printout of hundreds of pages). But the simple answer is, "No." – David K Feb 10 '15 at 14:06
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@DavidK can we make an antideravtive for functions like e^(-x²) and cos(x²) ... ? because a friend of mine says that if every continuous function has an antiderivative then we could make antiderivatives for those functions the same way we did for the antiderivative of 1/x – Mehdi Zibout Feb 10 '15 at 14:13
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The antiderivative exists, of course. It is $\int e^{-x^2};dx.$ But there is not an equal expression involving just elementary functions and no $\int$ symbol the way there is for, say, $\int x^2;dx.$ – David K Feb 10 '15 at 14:18
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@DavidK thank you for your help. but how can I now close this question? – Mehdi Zibout Feb 10 '15 at 14:21
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"Closing" on this site means something different from what I think you want, but I have summarized some of the comments in a community wiki answer. You can accept that answer if you like, and the question will then be counted as "answered". – David K Feb 10 '15 at 14:55
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Algorithms exist (notably the Risch algorithm) to decide whether an indefinite integral (antiderivative) can be written as a "closed form" involving only elementary functions. The implementation of such an algorithm is extremely complex, however, and you cannot always guarantee that it will give a yes/no answer to the question of whether there is a closed-form solution of whatever integral you ask it to solve.
Methods to evaluate arbitrary definite integrals in closed form are even more difficult to exactly decide, since in addition to the obvious technique of evaluating the antiderivative at each end of the interval of integration (if the antiderivative can be written in closed form), there are additional methods that can be employed on certain intervals of integration (such as the evaluation of $\int_0^\infty e^{-x^2}\;dx$).
A formal proof that this particular integral cannot be solved is therefore not practical (at least not for a format such as this site). An informal argument, however, is that the integral $F(x) = \int_0^x e^{-t^2}\;dt$ is of immense interest to mathematicians, and it is very unlikely that interesting results such as a closed-form expression for $F(1)$ in terms of elementary functions would have gone unnoticed.
