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This is Brilliant Integration Contest - Season 1 (Part 2) as a continuation of the previous contest (Part 1). There is a major change in the rules of contest, so please ...

Based on this problem.

Define two positive integer sequences \(\{a_n\}\) and \(\{b_n\}\) be defined as \(a_1\ne b_1\), \(a_{n+1}=a_n+k\) and \(b_{n+1}=b_n+k\). These ...

\[ \int_0^1 \sqrt[3]{\frac{\big\{\frac1x\big\}}{1-\big\{\frac1x\big\}}}\frac{dx}{1-x} \]

If the closed form of the value of the integral above can be expressed as ...

\[\int_0^{\frac{\sqrt{2} }{2} } \frac{\arcsin x}{x} dx = \frac{a}{b}G +\frac{c}{d}\pi^k \ln(m),\]

where \(a,b,c,d,k,m\) are positive ...

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