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Evaluate $\lim\limits_{{n}\to{\infty}} \int_{n}^{n+1} \cos^2(x^2) \,dx$

I've tried using the half angle identity and the taylor series on the remaining $1/2 + \cos(2x^2)$ to prove the value is $1/2$, but I am out of ideas.

I've tried using the half angle identity and the taylor series on the remaining $1/2 + \cos(2x^2)$ to prove the value is $1/2$, but I am out of ideas.

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