If the given integration is correct the derivative of

(u/8)*(2u^2 - a^2)sqrt( a^2 - u^2) + (a^4/8)arc sin (u/a) + C

should be u^2*sqrt ( a^2 - u^2)

[(u/8)*(2u^2 - a^2)sqrt( a^2 - u^2) + (a^4/8)arc sin (u/a) + C]'

=> [(u/8)*(2u^2 - a^2)sqrt( a^2 - u^2)]' + [(a^4/8)arc sin (u/a)]'

=> [(u^3/4 - u*a^2/8)sqrt( a^2 - u^2)]' + [(a^4/8)arc sin (u/a)]'

=> [((3/4)u^2 - a^2/8)*sqrt (a^2 - u^2) + (u^3/4 - u*a^2/8)*(-u)/sqrt (a^2 - u^2) + (a^4/8)/sqrt (a^2 - u^2)]

=>[((3/4)u^2 - a^2/8)(a^2 - u^2) + (u^3/4 - u*a^2/8)(-u) + (a^4/8)]/sqrt (a^2 - u^2)

=> [3u^2a^2/4 - a^4/8 - 3u^4/4 + u^2a^2/8 - u^4/4 + u^2a^2/8 + a^4/8]/ sqrt (a^2 - u^2)

=> [u^2a^2 - u^4]/ sqrt (a^2 - u^2)

=> u^2(a^2 - u^2)/sqrt (a^2 - u^2)

=> u^2 sqrt (a^2 - u^2)

**Therefore the given integration is correct.**