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Calculus Level 5

I = 0 1 0 x x 4 ln ( s ) ln ( 1 x ) d s d x \large I=\int _0^1 \int _0^x \frac {x^4}{\ln(s) \sqrt {\ln\left(\frac { 1 }{ x } \right)}} \ ds\ dx If I I can be expressed as 2 π A ln ( A + M ) \displaystyle -2\sqrt { \frac \pi A} \ln(\sqrt A +\sqrt M) , where A A and M M are square-free, find the 3-digit integer M A A \overline{MAA} .


The answer is 655.

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