So I need to figure out how to take the Mellin transform of
$$ f(x)=\int_2^x \sin(A_1/2)+\sinh(A_1/2)dt,$$
where $A_1=1/\ln(t).$ I'd also like to know how well the Mellin transform of $f(x)$ approximates $\frac{\log(s-1)}{s}.$
The Mellin transform is defined as:
$$ \{Mf\}(s)=\phi(s)=\int_0^{\infty} x^{s-1}f(x) dx, $$
and is an important integral transform in number theory.
I would like to find the integral transform of $f(x)$ because $f(x)$ approximates the prime counting function, $\pi(x)$ and computing this transform would help me take the next step in my studies.
I don't really know where to start because this is the first integral transform I've had to compute. If someone could clearly explain how to compute this transform it would be much appreciated.