Is there anything can not be integrated or differentiated?

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?
Thank you. :)

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.

Can I see some functions that can not be integrated? He likes to ask me questions and if i dont know the answer,he likes to stare into my soul with this look...... :S

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.

There are functions such that there is a function such that but yet is not integrable.

In other words, there are derivatives that have no integral.