Difficult Integration

I have a question that wants the definite integral between 0 and 1/2 of:

e^(x^2)*cos(x^2)

I'm know the basics of integration i.e. the definite integral between a and b = F(b) - F(a).
that's easier with e^x etc or e^ax

The problem is I don't even know where to start with this one!

Thanks

I agree that using something like Maple or Mathematica may not be satisfying, and though I don't know anything about quadrature method of numerical analysis, I believe this can be (pretty) accurately calculated using Trapezoidal Rule - or better yet, if you've learned it already - Simpson's Rule. I think that's the best way to approach it with basic knowledge of integration. It's just tedious.

I can honestly tell you that there is no way to express the antiderivative of this function in terms of elementary functions, or a finite number of roots of elementary functions. It is a bit complicated to argue this, but it is a result of Liouville Theory that this will not be expressible in terms of elementary functions, and so you're stuck with using some sort of numerical computation. The best I can do is 0.54, and this is a rough approximation based on the quadrature method of numerical analysis. The best that can be done analytically is in terms of the error function, which is dependent on the integral of e^(x^2), and the previous answer is exactly the output of Maple, Mathematica, etc. But again, this is unsatisfying, because one would like to be able to carry out a computation explicitly for this sort of question.

If there was even an x in front of all of this, then we could do this integral completely analytically, without having to fall back on numerical integration.

int(exp(x^2)*cos(x^2), x = 0 .. 1/2) = (1/4)*sqrt(π)*(erf((1/2)*sqrt( -1- i))*sqrt( - 1 + i) + erf((1/2)*sqrt( - 1 +i))*sqrt( - 1 - i))/(sqrt( - 1 - i)*sqrt( - 1 + i))