Deceptively difficult integral

Calculus Level 5

\[\overline { aaaaaaaa } \int _{ 0 }^{ 1 }{ { x }^{ 2 }{ \left( \int _{ 1 }^{ x }{ \frac { \sqrt { 10 } }{ t } dt } \right) }^{ 4 } } dx+1=\overline { abcdefghij }\] If \(\left\{ a,b,c,d,e,f,g,h,i,j \right\}\) is some permutation of all the single-digit, non-negative integers, find \(a\).

Details and Assumptions

The letters under the bar do not represent a product. Instead, each letter represents a distinct digit of the number represented.


This problem is original.

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