Suppose we define a piecewise function, frankensine(x) as described below.
frankensine(x)=⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧sin(x) ,0<x<2πsin(2x),2π<x<2π(1+21)sin(3x),2π(1+21)<x<2π(1+21+31)sin(4x),2π(1+21+31)<x<2π(1+21+31+41)⋮sin(nx),2π(1+21+31+…+n−11)<x<2π(1+21+31+41+…+n1)⋮
For integer n, let A=n→∞lim∫0nfrankensine(x)dx.
What is the value of ⌊1000A⌋?
Inspiration.