Approximating this integral without using Mertens' theorem Take $p$ as prime, $\text{li}(x)$ as logarithmic integral  and
$$
R(x)=\sum_{p\leq x}\frac{\ln p}p-\ln x
$$
Without using Mertens' theorem find
$$
\int_0^x\frac{tR'(t)}{\ln t}dt
$$
I tried using integration by parts and got stuck here:
$$
\frac{xR(x)}{\ln x}-\int_0^x\sum_{p\leq t}\frac{\ln p}p\left(\frac1{\ln t}-\frac1{\ln^2 t}\right)dt+x-\text{li}(x)
$$
where that integral is the term I can't solve!
An approximate solution is also welcome!
 A: Note that 
$$
\sum_{p\leq t}\frac{\ln p}{p}
$$
remains constant for $t$ between two successive primes. So if we write all the primes smaller than $x$:
$$
p_1 \leq p_2 \leq\ldots\leq p_n\leq x,
$$
then we can split up the integral into intervals:
$$
\begin{multline}
\frac{xR(x)}{\ln x}- \sum_{k=1}^{n-1}\sum_{j=1}^k\frac{\ln p_j}{p_j}\int_{p_k}^{p_{k+1}}\left(\frac1{\ln t}-\frac1{\ln^2 t}\right)dt\\
- \sum_{j=1}^n\frac{\ln p_j}{p_j}\int_{p_n}^{x}\left(\frac1{\ln t}-\frac1{\ln^2 t}\right)dt
+ x-\text{li}(x),
\end{multline}
$$
which is
$$
\begin{multline}
\frac{xR(x)}{\ln x}- \sum_{k=1}^{n-1}\sum_{j=1}^k\frac{\ln p_j}{p_j}\left(\frac{p_{k+1}}{\ln p_{k+1}} - \frac{p_{k}}{\ln p_{k}}\right)\\
- \sum_{j=1}^n\frac{\ln p_j}{p_j}\left(\frac{x}{\ln x} - \frac{p_{n}}{\ln p_{n}}\right) + x -\text{li}(x),
\end{multline}
$$
and it reduces to
$$
\begin{multline}
- \sum_{k=1}^{n-1}\sum_{j=1}^k\frac{\ln p_j}{p_j}\left(\frac{p_{k+1}}{\ln p_{k+1}} - \frac{p_{k}}{\ln p_{k}}\right)
+ \sum_{j=1}^n\frac{\ln p_j}{p_j}\frac{p_{n}}{\ln p_{n}} -\text{li}(x).
\end{multline}
$$
