Deceptively difficult integral

$\overline { aaaaaaaa } \int _{ 0 }^{ 1 }{ { x }^{ 2 }{ \left( \int _{ 1 }^{ x }{ \frac { \sqrt { 10 } }{ t } dt } \right) }^{ 4 } } dx+1=\overline { abcdefghij }$ 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.