Listing out all the integers

Calculus Level 4

$\large \int_1^\infty \frac { \{ x\} } {x^3} \, dx$

If the integral is equal to $$1- \dfrac{\pi^2}{A }$$, find the value of $$A$$.

Clarification

$$\{ x\}$$ denotes the the fractional part of $$x$$. For instance, $$\{2\} = 0, \{3.4\} = 0.4, \{\pi\} = \pi - 3$$.

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