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- #1

$$

\int_0^{\infty}\frac{x^2}{x^6 + 1}dx = \frac{\pi}{6}

$$

Can I do this:

$$

\int_0^{\infty}\frac{x^2}{x^6 + 1}dx = \frac{1}{2}\int_{-\infty}^{\infty}\frac{x^2}{x^6 + 1}dx

$$

and solve the integral like this

$$

\int_{-\infty}^{\infty}\frac{x^2}{x^4 + 1}dx = 2i\pi\sum_{z \ \text{upper half}}\text{Res}_{z}f = \frac{\pi\sqrt{2}}{2}

$$