We need to evaluate the definite integral of the function $f(x) = \frac{1}{x}$ over the interval $[-5, -1]$. The integral is given by: $$I = \int_{-5}^{-1} \frac{1}{x} dx$$
The antiderivative of $\frac{1}{x}$ is $\ln|x|$. Applying the limits from $-5$ to $-1$: $$I = [\ln|x|]_{-5}^{-1}$$
Substitute the upper and lower limits into the antiderivative: $$I = \ln|-1| - \ln|-5|$$ $$I = \ln(1) - \ln(5)$$
Since $\ln(1) = 0$, the expression simplifies to: $$I = 0 - \ln(5) = -\ln(5)$$
Final Answer: -log 5
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