Listing out all the integers

Calculus Level 4

1{x}x3dx \large \int_1^\infty \frac { \{ x\} } {x^3} \, dx

If the integral is equal to 1π2A 1- \dfrac{\pi^2}{A } , find the value of AA.

Clarification

{x} \{ x\} denotes the the fractional part of xx. For instance, {2}=0,{3.4}=0.4,{π}=π3 \{2\} = 0, \{3.4\} = 0.4, \{\pi\} = \pi - 3 .

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