Grandiose Gibberish

Calculus Level 4

Suppose we define a piecewise function, frankensine(x) \text{frankensine}(x) as described below.

frankensine(x)={sin(x) ,0<x<2πsin(2x),2π<x<2π(1+12)sin(3x),2π(1+12)<x<2π(1+12+13)sin(4x),2π(1+12+13)<x<2π(1+12+13+14)sin(nx),2π(1+12+13++1n1)<x<2π(1+12+13+14++1n) \text{frankensine}(x) = \begin{cases} \sin(x) \ \quad, \quad 0 < x < 2\pi \\ \sin(2x) \quad, \quad 2\pi < x < 2\pi\left(1 + \frac12 \right) \\ \sin(3x) \quad, \quad 2\pi\left(1 + \frac12 \right) < x < 2\pi\left(1 + \frac12 + \frac13\right) \\ \sin(4x) \quad, \quad 2\pi\left(1 + \frac12 + \frac13\right) < x < 2\pi\left(1 + \frac12 + \frac13+\frac14\right) \\ \vdots \\ \sin(nx) \quad, \quad 2\pi\left(1 + \frac12 + \frac13+\ldots + \frac1{n-1}\right) < x < 2\pi\left(1 + \frac12 + \frac13+\frac14+\ldots + \frac1n \right) \\ \vdots \end{cases}

For integer nn, let A=limn0nfrankensine(x)dx. \displaystyle A = \lim_{n\to\infty} \int_0^n \text{frankensine}(x) \, dx.

What is the value of 1000A \lfloor 1000A \rfloor ?


Inspiration.

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