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 $$ \begin{equation*} \int_{-\infty}^{\infty} \frac{dk}{k-k_0}= \int_{\infty}^{0} \frac{-du}{-u-k_0}+ \int_{0}^{\infty} \frac{dk}{k-k_0}= \int_{0}^{\infty} \frac{dk}{-k-k_0}+ \int_{0}^{\infty} \frac{dk}{k-k_0}=0. \end{equation*} $$