I would like to know the value of the following integral:
$$\displaystyle\int_2^\infty ({\theta(y)-y})\frac{\mathrm d}{\mathrm dy}\frac{\ln(y-1)}{\ln(y)} \mathrm dy$$
(Where $\theta(y)$ is Chebyshev's First Function)
It appears on (4.16) of J. Barkley and L. Schoenfeld "Approximated Formulas for some functions of Prime Numbers".
I do not know where to start from, and neither Mathematica nor Mathlab seem to help.
How can I find at least an approximated value for it?
Might be helpful. You have an integral of the form
$$\int f g'\ \text{d} y$$
Use by parts integration to get
$$\underbrace{\left(\theta(y) - y)\right)\frac{\ln(y-1)}{\ln(y)}\bigg|_2^{+\infty}}_{0} - \int_2^{+\infty} \left(\theta(y) - y\right)'\ \frac{\ln(y-1)}{\ln(y)}\ \text{d}y$$
Now
$$\frac{\text{d}}{\text{d}y} \theta(y) = \sum_{p\leq y} \frac{1}{p}$$
$$-\left(\sum_{p\leq y} \frac{1}{p} - 1\right)\int_2^{+\infty} \frac{\ln(y-1)}{\ln(y)}\ \text{d}y$$
Your problem arises when you try to calculate the last integral, which does diverge.
We can approach it with a consideration like at infinity the integral approaches to the value $1$.
I think now it's your turn to make some approximation to that integral.