Compute the integral: $$\int_0^{\infty} \frac{x^p}{x^2 + 2x\cos\lambda+1}dx, (-1 < p < 1, -\pi < \lambda < \pi)$$ with complex analysis techinques.

What I've tried in this problem (because the problem is located among other problems that I've solved using the $\textit{circle that avoids positive part of x-axes}$) is to find residues in the singularities and that would give me solution. But, in case of $0 < p < 1$ this integral does not have same properties as in case $- 1 < p < 0$ (we had classified them as separate types), so I am clueless here. Any help is appreciated. Should I use any particular curve that is non-standard, or I am missing something? Thank you.

$\textbf{Edit:}$ Also, it is clear to me that the $p$ is chosen in such way that the integral converges.