# Math Help - Fresnel Integral

1. ## Fresnel Integral

Evaluate $\int_0^\infty\cos x^2\,dx$

2. Originally Posted by liyi
Evaluate $\int_0^\infty\cos x^2\,dx$

3. Originally Posted by liyi
Evaluate $\int_0^\infty\cos x^2\,dx$
$\sqrt{\frac{\pi}{8}}$ if I remember correctly. Too lazy to do this integral because I done it before.

4. Originally Posted by liyi
Evaluate $\int_0^\infty\cos x^2\,dx$
Substitute $u=x^2,$ the integral becomes to $\int_0^\infty {\frac{{\cos u}}{{2\sqrt u }}\,du} .$

The following parameter may be useful: $\int_0^\infty {\frac{{e^{ - \alpha u} }}{{\sqrt \alpha }}\,d\alpha } = \frac{{\sqrt \pi }}{{\sqrt u }}.$

To prove this, we can use a substitution and the remarkable fact $\int_0^\infty {e^{ - x^2 } \,dx} = \frac{{\sqrt \pi }}{2}.$

Now plug the parameter into the integral, so we'll have a double integral which can be computed without problems.