which is a standard integral.
You don't need any kind of u or whatever conversion. It is a simple basic integral. All you need to know is:
int(1/t) = ln(t). THAT'S IT.
1/2 is just a constant. So the answer of your questions is:
int(1/2t)=(1/2) int(1/t)=(1/2) ln(t). AND THAT'S IT.
Which may be different than ln(t/2).
a*ln(t)=ln(a*t) is NOT always true.
So (1/2)*ln(t) = ln((1/2)*t) is NOT always true.
If for some reason you decided to use a u-substitition:
We then get:
Looks a bit different. But recall:
But is nothing more than a constant. So we'll combine it with to make a new constant of integration which we'll call .
In other words, and are two legit antiderivatives of