Answer: E

Small Hint:

Copy arc $\overset{\frown}{CA}$ to an arc $\overset{\frown}{MN}$ ending at a new point $N$ between $M$ and $B$

Big Hint:

Drop $NQ\perp AB$; use equal arcs to identify a rectangle and equal horizontal end segments

Solution:

Choose $N$ on arc $\overset{\frown}{MB}$ so that arcs $\overset{\frown}{MN}$ and $\overset{\frown}{CA}$ are equal, and drop $NQ\perp AB.$ Then $MN=AC=x.$ The remaining equal arcs $\overset{\frown}{AM}$ and $\overset{\frown}{NB}$ give equal corresponding chord projections, so $QB=AP=x+1.$ Also $MNPQ$ is a rectangle, hence $PQ=MN=x.$ Therefore $$\begin{aligned} PB&=PQ+QB\\ &=x+(x+1)\\ &=2x+1. \end{aligned}$$

Therefore, the correct answer is E.