In Multiplicative Number Theory, Montgomery and Vaughan provide an identity for Riemann's zeta function.
But in step 1 I am not clear how they separated the summation, in step 2 I am completely lost at how they reached those integrals (that step is the one that gives me the most trouble) and in step 3 I don't know how they integrated by parts. I need some clarification. It seems to me they didn't write several steps.
For the first step, the first the summation is from $1$ to $\lfloor x\rfloor$ and the other one is from $\lfloor x\rfloor+1$ to $+\infty$. For the second step, $\lfloor u\rfloor=u-\{u\}$ therefore $d\lfloor u\rfloor=du-d\{u\}$. As for step 3, the integration by parts gives that $$ \int_x^{+\infty}u^{-s}d\{u\}=\left[u^{-s}\{u\}\right]_x^{+\infty}-\int_x^{+\infty}\{u\}d(u^{-s})=x^{-s}\{x\}+s\int_x^{+\infty}\{u\}u^{-s-1}du $$ And the integral $\int_x^{+\infty}u^{-s}du$ can easily be computed, its value is $\frac{x^{1-s}}{s-1}$ so at the end you get the result.