Integration Of exponential Function I have tried almost everything, but can't solve this integral.
$$\int e^{-1/x^2} \, dx $$

To some extent, you have in fact given the answer yourself, if you have (literally) tried everything then you have proven it is impossible to integrate in terms of elementary functions. This is indeed the case.
This statement is similar to the impossibility of solving a polynomial of degree $\geq$ 5 in terms of radicals; also the impossibility of "squaring the circle", namely constructing (in a finite number of steps) a square with straight edge and compass which has the same area of a given circle; also the impossibility of trisecting a given angle with just a straight edge and compass. The proofs of these statements can be learned through the topic of Galois theory. 
