Complex integral and contour

Question

I'm trying to compute the integral $\int\limits_0^\infty \frac{dx}{2+x^n}$

So my attempt is finding the roots of $2+x^n$ that are $e^{\pi i}$$e^{\frac{2 \pi m}n} $ for $m=1,2...,n-1$ and then define the contour of the integral.

After that apply Residue Theorem.

Please tell me if I'm doing this right.

Answer 1

Not a full solution (so please don't downvote).

Here are your poles: