1) $\displaystyle \int_{0}^{\infty} e^{-ax^{2}} \sin \Big( \frac{b}{x^{2}} \Big) \ dx $

2) $\displaystyle \int_{0}^{\infty} e^{-ax^{2}} \cos \Big(\frac{b}{x^{2}} \Big) \ dx $

3) $\displaystyle \int_{0}^{\infty} \cos ax^{2} \cos 2bx \ dx $

4) $\displaystyle \int_{0}^{\infty} \sin ax^{2}} \cos 2bx \ dx $

I can evaluate all of these integrals, but using a method that's just too difficult to justify. Namely evaluating an integral with real-valued parameters and assuming the solution is valid when one of the parameters is imaginary.