First of all, what is the elementary methods for solution of integrals?Who decides this?For example,Evaluate $\int x^2e^{x^2} dx$ and $\displaystyle\sqrt{lnx}dx$ or $\displaystyle\int e^{sin(e^x)}dx$
When I try to solve these integrals and fail,then want wolfram to solve these,and I always get this
$$\text{(no result found in terms of standard mathematical functions)}$$
What is the "standard mathematical function", Can $\left[\dfrac2{\sqrt \pi}\displaystyle\int^x_0 e^{t^2}dt\right]$ be a standard function?Why not?
What is the tiny line between standard and non-standard functions?Why we can't solve these non-elementary integrals with elementary methods?Can we define or fidn all elementary methods?
