As Little Work As Possible

Calculus Level 5

A particle can move on an infinite two-dimensional plane, which is given an \(Oxy\)-coordinate system. As the particle moves on the plane, when it is at the point with coordinates \((x,y)\), it experiences a resistance force equal to \(e^{-y}\) Newtons in a direction opposite to that of motion of the particle. Thus the work done moving the particle along a particular path \(C\) is \[ \int_C e^{-y}\,ds \] Joules, where \(s\) represents the arc-length, measured in metres.

The particle has to be moved smoothly from the point \(A=(0,0)\) to the point \(B=\big(\tfrac12\pi,1\big)\), in such a way that the \(x\)-coordinate of its position never decreases. Thus a possible path from \(A\) to \(B\) is of the form \((x,y(x))\), where \(y\,:\, \left[0,\tfrac12\pi\right] \to \mathbb{R}\) is a smooth function such that \(y(0) = 0\) and \(y\big(\tfrac12\pi\big) = 1\).

Of all the possible paths from \(A\) to \(B\) there is one for which the work done against the resistance force in moving the particle from \(A\) to \(B\) is as small as possible. The length of this path can be shown to be \[ P \tanh^{-1}\left( \frac{\sqrt{e^Q + R}}{Se + T}\right), \] where \(P,Q,R,S,T\) are positive integers. What is the value of \(P+Q+R+S+T\)?

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