What is the value of $\int^{1}_{-1}\left(\frac{\sin^{-1}{x}}{\sqrt{2-x^{12}}}+|x|\right)\,dx?$

$\frac{2\pi}{9}$
$\frac{2\pi}{9}+1$
$\frac{\pi}{9}+\frac{1}{2}$
1

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$\frac{\sin^{-1}{x}}{\sqrt{2}-x^{12}}$ is an odd function, thus, the interval $[-1,1]$ is symmetric about 0. This means the first part of the integral is 0.

Now, since $|x|$ is even, $\int^{1}_{-1}|x|\,dx=2\int^{1}_{0}x\,dx=1$