I want to compute the fourier transform of the function $\cos(2πt^2)$
But when faced to this integral : $$ \int_{-\infty}^{\infty} \cos(2\pi t^2)e^{-i2πft}dt = \int_{-\infty}^{\infty} \dfrac{e^{i2πt^2}+e^{-i2πt^2}}2e^{-i2πft}dt $$
I don't see how reducing $e^{i2πt^2}e^{-i2πft}$ to $e^{i2πt^2-i2πft}$ helps in solving the integral since we still don't have something factored by t. I think i'm missing a step in my reasoning.
Thanks
With this kind of indication, I suppose you are just expected to complete the square. For example for one of the two terms it gives $$ \int e^{-2i\pi t^2}\,e^{-2i\pi xt}\,\mathrm d t = e^{i\pi x^2/2}\int e^{-2i\pi (t+x/2)^2}\mathrm d t = e^{i\pi x^2/2}\int e^{-2i\pi t^2}\mathrm d t = \frac{e^{i\pi x^2/2}}{2\,(1+i)}. $$