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So, there are many ways to calculate the integral $\int_{-\infty}^{+\infty} \sin(x^2)\,\mathrm{d}x$. For example, you can use differentiation under the integral sign or you can use complex numbers. My teacher asked me to find the value of this Fresnel integral using double improper integral. Perhaps one may use $\iint\limits_\mathbb{R^2} \sin(x^2 + y^2)\,\mathrm{d}x\,\mathrm{d}y$? But I don’t know yet how this works here.

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