Integrate By Parts Or Use The Beta Function

Calculus Level 4

\[I=\dfrac{\displaystyle \int_0^1 (1-x^{1729})^{1729} \, dx}{\displaystyle \int_0^1 (1-x^{1729})^{1728} \, dx}\]

If \(I\) can be represented as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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