# Integrating Chebyshev theta function

I'm trying to compute the following integral ($\vartheta(x) = \sum\limits_{p \leq x}\log(p)$)

$$\int\limits_{0}^{\infty}\vartheta(e^x) e^{-(1+s)x} \text{dx}$$

The result is supposed to be $\frac{\sum\log(p)/p^{s+1}}{1+s}$, but I'm having trouble doing it. I tried to substitute $t=e^x, dt/t=dx$ and I got

$$\int\limits_{1}^{\infty} \vartheta(t)t^{-(2+s)} \text{dx}$$

And I'm stuck. I don't know how to evaluate an integral with a sum inside that depends on in

where $[\,\cdot\,]$ is the Iverson bracket, $[x \geqslant \log p]$ is $1$ if $x\geqslant \log p$, and $0$ if $x < \log p$.