Question on $\sum_{p\leq x}f(p)$

Question

In the paper of J. Barkley Rosser and Lowell Schoenfeld http://www.seanerikoconnor.freeservers.com/Mathematics/AbstractAlgebra/PrimitivePolynomials/Approximate_Formulas_for_Some_Functions_of_Prime_Numbers.pdf page 68, they obtained the following formula

My question is : should we assume that the integral 2.28 is convergent or this integral is convergent by proof?

Answer 1

Your screen-shot cut off half of the explicit answer to your question (emphasis by me):

If the integral in (2.28) below con-

verges, we can rewrite (2.26) as