Let $u = 2 \pi x^2$. Then, $du = (4 \pi x) dx$.
Rearranging the equation, we have $dx = \frac{du}{4 \pi x}$.
Substituting these values into the integral, we get:
$$\int x \cos(2 \pi x^2) dx = \int \frac{1}{4 \pi} \cos(u) \frac{du}{x}$$
Simplifying, we
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