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Problem originally posted in a technical math section, so missing the template.

So i am trying to find the ##\int_{0}^{\infty} cos(x^2) dx##. I used Eulers identity to get ##\int_{0}^{\infty} cos(x^2) - isin(x^2) dx = \int_{0}^{\infty} e^{-i(x^2)} dx##. I squared this integral, changed to polar and evaluated and at the end of this process i got the result of ##\frac{1}{2} \sqrt{\frac{\pi}{i}}##. Where do i go from here? Am i even on the right track? Thanks for your help.