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\int_{0}^{\infty}\sin{t^{2}} dt=\sum_{n=0}^{\infty}\int_{\sqrt{nx}}^{\sqrt{(n+1)x}}\sin(t^{2}) dt \tea{gilt da:}\tag{*}\\ \int_{0}^{\infty}t^{2} dt=\int_{\sqrt{t}}^{\sqrt{t+1}}t^{2} dt +\int_{\sqrt{2t}}^{\sqrt{2(t+1)}}t^{2}dt\dots\\ *=\sum_{n=0}^{\infty} \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}}(-1)^{n}...