does Integrating both sides of an equation in dx will Invalidates the equality?

I'm struggling to grasp the justification behind integrating both sides of an equation. While I understand that operations can be applied to both sides, maintaining equality, it appears that this principle doesn't apply here.

given the equality $x=y$ integrating both sides by dx would give $\frac{x^2}{2} = xy$

but this seems not to be valid since if I start from x=y=5 , I would get $\frac{25}{2}=25 $that's not true.

given for instance $log(a)=log(b)$if I integrate both sides by $da$I get:$alog(a)-a=log(b)a$

if $a=b=2$ then

$alog(a)-a=-0.61..$and

$log(b)a=1.38$

that are different, what I'm doing wrong here?