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

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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