<s>[INST] Show that $\int_0^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx=\alpha \ln\alpha$ 
Show that the improper integral $\int_0^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx=\alpha \ln\alpha$, for $\alpha\in(0,1)$.

This is an integral of Riemann. My work:


*

*The set of discontinuities of the integral is


$$D=\left\{\frac1k:k\in\Bbb N\right\}\cup\left\{\frac{\alpha}{k}:k\in\Bbb N\right\}$$


*And when we have that $x>\alpha$ the integral can be simplified to


$$\int_0^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx=\int_0^\alpha \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx-\alpha\int_{\alpha}^1 \left\lfloor\frac{1}{x}\right\rfloor\mathrm dx$$
I dont know how to continue from here, it is not clear how to handle the partition $D$ to simplify the integral. What I did here is just see what is the value of $\int_{\alpha}^1 \left\lfloor\frac{1}{x}\right\rfloor\mathrm dx$ to see if I get some clue.
If there is no weird mistake somewhere:
$$\int_{\alpha}^1 \left\lfloor\frac{1}{x}\right\rfloor\mathrm dx=\int_\alpha^{\frac1{\left\lfloor 1/\alpha\right\rfloor}}\frac{\mathbf 1_{\Bbb N}(1/\alpha)\mathrm dx}{\lfloor 1/\alpha\rfloor}+\sum_{k=1}^{\lfloor1/\alpha\rfloor}\int_{\frac1{k+1}}^{\frac1k}\frac{\mathrm dx}{k}=\\=\mathbf 1_{\Bbb N}(1/\alpha)\frac{1-\alpha\lfloor 1/\alpha\rfloor}{\lfloor 1/\alpha\rfloor^2}+\sum_{k=1}^{\lfloor1/\alpha\rfloor}\frac1{k^2(k+1)}$$
what is not useful at all. So I get stuck with this problem, can you help me to show this identity (not going deeper than a Riemann integral background)? Thank you in advance.
 [/INST] The improper integral can be evaluated as the limit of an integral over $[1/n,1]$ as $n \to \infty.$
Making the change of variables $u = 1/x,$ we get 
$$\alpha\int_{1/n}^1 \left\lfloor\frac{1}{x}\right\rfloor dx = \alpha\int_{1}^{n } \frac{\left\lfloor u\right\rfloor}{u^2} \, du  \\ = \alpha \sum_{k=1}^{n-1}\int_{k}^{k+1 } \frac{\left\lfloor u\right\rfloor}{u^2} \, du \\ = \alpha \sum_{k=1}^{n-1}\int_{k}^{k+1 } \frac{k}{u^2} \, du  \\ = \alpha \sum_{k=1}^{n-1} k\left(\frac{1}{k} - \frac{1}{k+1} \right) \\ = \alpha \sum_{k=2}^{n} \frac{1}{k},\tag{1}$$
and, making the change of variables $u = \alpha/x,$ we get
$$\int_{1/n}^1 \left\lfloor\frac{\alpha}{x}\right\rfloor dx= \alpha \int_{\alpha}^{n \alpha} \frac{\left\lfloor u\right\rfloor}{u^2} \, du \\ = \alpha \int_{\alpha}^{n} \frac{\left\lfloor u\right\rfloor}{u^2} \, du  - \alpha \int_{n \alpha}^{n} \frac{\left\lfloor u\right\rfloor}{u^2} \, du \\ = \alpha \int_{1}^{n} \frac{\left\lfloor u\right\rfloor}{u^2} \, du  - \alpha \int_{n \alpha}^{n } \frac{\left\lfloor u\right\rfloor}{u^2} \, du\tag{2}$$
Subtracting (1) from (2) we eliminate the divergent harmonic sum and obtain
$$\int_{1/n}^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor - \alpha \left\lfloor\frac{1}{x}\right\rfloor \right) dx = - \alpha \int_{n \alpha}^{n} \frac{\left\lfloor u\right\rfloor}{u^2} \, du \\ = - \alpha \int_{1}^{n} \frac{\left\lfloor u\right\rfloor}{u^2} \, du + \alpha \int_{1}^{\lfloor n \alpha \rfloor} \frac{\left\lfloor u\right\rfloor}{u^2} \, du + \alpha \int_{\lfloor n \alpha \rfloor}^{n \alpha} \frac{\left\lfloor u\right\rfloor}{u^2} \, du \\ = -\alpha \sum_{k=2}^{n} \frac{1}{k} + \alpha \sum_{k=2}^{\lfloor n \alpha \rfloor}\frac{1}{k} + \alpha \frac{n \alpha - \lfloor n \alpha \rfloor}{n \alpha} \\ = -\alpha \left(\sum_{k=1}^{n} \frac{1}{k} - \log n \right)  + \alpha \left( \sum_{k=1}^{\lfloor n \alpha \rfloor}\frac{1}{k} - \log\lfloor n \alpha \rfloor \right) - \alpha \log n + \alpha \log \lfloor n \alpha \rfloor  + \alpha \frac{n \alpha - \lfloor n \alpha \rfloor}{n \alpha}$$
From here, it should be relatively straightforward to take the limit as $ n \to \infty$ to obtain $\alpha \log \alpha$.
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