We can repeat the above subtraction trick for more complicated integrands. First we modify the integration limits to $\pm \infty$ and use the fact that $$\begin{equation*} \int_{-\infty}^{\infty} \frac{dk}{k-k_0}= \int_{-\infty}^{0} \frac{dk}{k-k_0}+ \int_{0}^{\infty} \frac{dk}{k-k_0} =0. \end{equation*}$$ A change of variable $u=-k$ in the integral with limits from $-\infty$ to $0$ gives $$\begin{equation*} \int_{-\infty}^{\infty} \frac{dk}{k-k_0}= \int_{\infty}^{0} \frac{-du}{-u-k_0}+ \int_{0}^{\infty} \frac{dk}{k-k_0}= \int_{0}^{\infty} \frac{dk}{-k-k_0}+ \int_{0}^{\infty} \frac{dk}{k-k_0}=0. \end{equation*}$$