What is the change of variable for proving the fractional part integral?

[alyafey22]

Prove the following

\(\displaystyle \int^1_0 \left \{ \frac{1}{x} \right \} \, dx = 1- \gamma \)​

 

[chisigma]

Prove the following

\(\displaystyle \int^1_0 \left \{ \frac{1}{x} \right \} \, dx = 1- \gamma \)​

[sp]With the change of variable $\displaystyle \frac{1}{x}= t $ the integral becomes...

$\displaystyle \int_{0}^{1} \{\frac{1}{x}\}\ d x = \int_{1}^{\infty} \frac{\{t\}}{t^{2}}\ d t = \sum_{n=2}^{\infty} \int_{n-1}^{n} (\frac{1}{t} - \frac{n-1}{t^{2}})\ d t = \lim_{n \rightarrow \infty} (\ln n - \sum_{k=2}^{n} \frac{1}{k})= 1 - \gamma$[/sp]

Kind regards

$\chi$ $\sigma$