Question

Super hard calculus problem! Can you help?
The closed form solution of the integral of e^(x^2)dx

Answer 1

There is no closed form solution, however the integral of e^(x^2) can easily be found using the maclaurin expansion for e^u.

e^u = 1+ u +u^2/2! + u^3/3! + u^4 /4! +....

let u=x^2

e^(x^2) = 1 + x^2 + x^4/2! + x^6/3! + x^8/ 4!+.....

Then integral of e^(x^2)dx is

x + x^3/ 3 + x^5/(5*2!) + x^7/(7*3!) + x^9/(9*4!) + ...... + C

That is its exact antiderivitive (and not an approximation).

No 'closed form' antiderivitive means its antiderivitive cannot be expressed as a sum,product,difference, or quotient of a finite number of elementary functions. Elementary functions are typically considered to be the trigonometric, inverse trigonometric, exponential, logarithmic, and algebraic functions.

Answer 2

Nothing hard about it. It can't be done. Look up "erf(x)" or the Error Function. There is plenty of literature on it.