is a continuous function. Therefore it is integrable. It's integral cannot however be expressed in terms of elementary functions.
There are lots of functions that cannot be integrated at all.
My teacher was saying that you can not integrate e^x^2(with no limits).He did say something can not be integrated but then later in the course it turned out you can do it by parts.So Is this true?or there is a way that he is not telling me?
I assume we're talking about Riemann integration here. Any function that can't be integrated must be messy because all piecewise continuous functions are integrable.
Here's a classic example: Let f(x) = 0 if x is rational, and f(x) =1 if x is irrational. Then the lower sums converge to 0 and the upper sums converge to 1, so that the integral does not exist.