Finding an alternative to a no closed form integral In my notes, I have learnt that some functions have no closed form integrals, for example
$$f(x) = e^{-x^2}$$ has no closed form integral.
I have 2 questions on this


*

*I understand that a closed form integral is a function which can be explicitly defined for , say $f(x)$. Is this true?

*If we have such a function like the above, how do you find the integral? I am not a student of complex numbers, so if it does involve complex numbers, please offer (perhaps) some explanation so I could research more.


Thank you.
 A: "Closed forms" aren't a particularly deep concept; they just mean that you've made a decision to consider certain operations special, and a closed form is just an expression written in terms of the special operations.
If you've decided to consider only $+, -, \cdot, \div$ as special, then simple things like $\log x$, $\sin x$, or even $x^y$ aren't closed forms!
On the other hand, if you include definite integration in your list, then the integral of any closed form is also automatically a closed form.

How does one find an integral? It depends on what you mean by "find". If you mean "write as a closed form expression", then simply can't find many integrals.
However, a more realistic meaning of "find" is to have some level of understanding of the integral. For example, for many applications, "finding" a function simply means that you have a way to compute numeric estimates of its values (i.e. given a decimal constant, be able to write another decimal constant that is approximately the value of the function), have a general idea of the overall shape of its graph, and/or have a closed form asymptotic expression (e.g. such a thing for $\sin x$ near $x=0$ is $x + O(x^3)$).
Methods to do these sorts of things are a major part of what you're learning in calculus.
A: There are numerical methods for finding good approximations to definite integrals $\int_a^bf(x)\,dx$ when there is no closed-form formula for the antiderivative $\int f(x)\,dx$. Perhaps you have a calculus book that discusses the trapezoid rule and methods of the nature. If not, you could do a websearch. 
