Problem Analysis #42

Consider the function$f(x) = \frac{1}{2}x^2 - x + \ln(1 + x),$and let $t$ be a positive real number. For $s > 0$, let $(s, f(s))$ be a point on the curve $y = f(x)$. Let $g(t)$ denote the value of $s$ such that the distance between the foot of the perpendicular from $(s, f(s))$ to the $y$-axis and the point where the tangent to the curve at $(s, f(s))$ meets the $y$-axis is equal to $t$. Evaluate$\int_{\frac{1}{2}}^{\frac{27}{4}} g(t) \, dt.$[4 points]\subsection*{Numerical answer}If the value is $\frac{p}{q} + \ln 2$, compute $p+q$.\