Need Major Help .. Integral?

Question

integral of e^(x^2)*cos(x^2) dx (0,1/2)

Please.. thanks

The trig part is giving me nighmares...would appreciate the help

Answer 1

Unfortunately , if you are trying to do this
by using an antiderivative, you are out
of luck. The integral is non-elementary.
Let's let u = x², x = √u, dx = du/2*√u
So(ignoring the limits for now), we get
½∫ e^u cos(u)/√u du
and the integral of this involves the imaginary error function.
You will either have to expand e^(x²)cos(x²) in
a series and integrate it term by term or use
a numerical integration procedure to compute it.
Edit:
Another way of seeing how your integral reduces
to the imaginary error function(if you don't
mind working with complex numbers) is
to use the identity
cos(x²) = ½(e^(ix²) + e^(-ix²) ).
A bit of easy manipulation will reduce it to 2 integrals of the
form K∫ e^(u²) du, where K is a (complex) constant.
Finally, ∫ e^(u²) du = ½√π*erfi(u) + C.
I'll let you work out the details!

Answer 2

This is a nonelementary integral, i.e. its integrand does not have elementary antiderivative. It requires numerical methods to be evaluated.

Answer 3

maybe use integration by parts?