Examples of Section 7.3 Integrals

Example with (Sqrt[(a^2) - (x^2)])

f  =   ∫(Sqrt[11 - 
                (x^2)]) \[DifferentialD]x

Simplify[(∂_x) f]

Example with (Sqrt[(a^2) + (x^2)])

f = ∫((Sqrt[11 + (x^2)])/(x^2)) \[DifferentialD]x

Simplify[(∂_x) f]

Example with (Sqrt[(x^2) - (a^2)])

f = ∫((Sqrt[(x^2) - 11])/((x^2) )) \[DifferentialD]x

Simplify[(∂_x) f]

Some funny Mathematica behavior

Clear[a]

    
∫(Sqrt[(a^2) - (x^2)]) \[DifferentialD]x

Assuming[
      a  >  0,   ∫(Sqrt[(a^2) - 
                (x^2)]) \[DifferentialD]x]

∫(Sqrt[9 - 
              (x^2)]) \[DifferentialD]x