Attached integral calculator cannot solve

[Werner]

You just have to realise that

'∫(-2,2,(X^3*COS(X/2)+1/2)*√(4-X^2),X)'
= '∫(-2,2,X^3*COS(X/2)*√(4-X^2),X)' + '∫(-2,2,1/2*√(4-X^2),X)'

The first part is an odd function, so the integral is zero, and we are left with the easy second part.

Cheers, Werner