You want e to the power x2, right?
The Wikipedia gives an answer, at http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions. I won't copy it here, too much trouble; and I don't know how to draw equations here.
This answer is quite complicated; it seems you basically use successive approximations to get the integral (antiderivative), instead of a simple expression as you might expect.
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I assume you mean -10x^4? In that case, antiderivative would be to add one to the exponent, then divide by the exponent. So -10x^5, then divide by 5. So the antiderivative is -2x^5.
By antiderivative do you mean integral? If yes, integral x^1 dx= (x^2)/2
(that weird integral or antiderivative sign) x^(-6/5) dx =-5*x^(-1/5)
It is an inverse function of a derivative, also known as an integral.
The general formula for powers doesn't work in this case, because there will be a zero in the denominator. The antiderivative of 1/x is ln(x), that is, the natural logarithm of x.