\[ \large \int\limits_{0}^{\infty} \frac{x^k}{1+x^n} \, dx = \, ? \]

for: \( k,n \in \mathbb{Z} \) and \( 0 \le k \le n-2 \)

EDIT: The The \( \pi \) s of the suggested answers in the denominator are supposed to be inside the \( sin( * ) \)

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