Вычисление собственных интегралов от параметра

Clear["Global`*"]
Limit[Integrate[Sqrt[x^2+a^2],{x,-1,1}],a->0]

Clear["Global`*"]
Limit[Integrate[x/y^2 Exp[-x^2/y^2],{x,0,1}],y->0]

Clear["Global`*"]
D[Integrate[Sin[x y]/x,{x,a+y,b+y}],y]
FunctionExpand[%]

Clear["Global`*"]
Integrate[Log[a^2 Sin[x]^2+b^2 Cos[x]^2],{x,0,Pi/2},Assumptions->{a>0,b>0}]