Assemble a continuous expectation using partial fractions
A continuous random variable X has the following density, with K chosen to make its total integral 1. Find K, then construct the mean integral and decompose its integrand in the supplied form. Report the exact expectation as P ln(R)+Q, using the stated endpoint ratio for R. All logarithms are natural
\[f(x)=\begin{cases}\dfrac{K}{(x+(8))^2},&-3\le x\le-1,\\0,&\text{otherwise},\end{cases}\]
\[\frac{x}{(x+(8))^2}=\frac{A}{x+(8)}+\frac{B}{(x+(8))^2},\quad E(X)=P\ln R+Q\]
For R use the value of x+(8) at the upper support endpoint divided by its value at the lower endpoint. This fixes the template; equivalent logarithmic forms exist