Answer
Divergent
Work Step by Step
\[\begin{align}
& \int_{-\infty }^{\infty }{\frac{2x+4}{{{x}^{2}}+4x+5}}dx \\
& \text{Therefore} \\
& \int_{-\infty }^{\infty }{\frac{2x+4}{{{x}^{2}}+4x+5}}dx=\int_{-\infty }^{0}{\frac{2x+4}{{{x}^{2}}+4x+5}}dx+\int_{0}^{\infty }{\frac{2x+4}{{{x}^{2}}+4x+5}}dx \\
& \text{Integrating} \\
& =\left[ \ln \left( {{x}^{2}}+4x+5 \right) \right]_{-\infty }^{0}+\left[ \ln \left( {{x}^{2}}+4x+5 \right) \right]_{0}^{\infty } \\
& =\left[ \ln \left( 5 \right)-\ln \left( \infty \right) \right]+\left[ \ln \left( \infty \right)-\ln \left( 5 \right) \right] \\
& =\infty \\
& \text{The improper integral diverges} \\
& \text{Divergent} \\
\end{align}\]
