# Mellin transform involving $\sinh({A_1}/2)$

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.

• Upon quick inspection, the first term in the integral defining the function $f$ seems to be diverging as $ln(x)$ as $x\rightarrow\infty$ and the second diverges exponentially. I don't believe you can Mellin transform this function... – DinosaurEgg Mar 9 at 2:48