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

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The discussion focuses on proving the integral $$\int^1_0 \left \{ \frac{1}{x} \right \} \, dx = 1 - \gamma$$ using a change of variable where $\frac{1}{x} = t$. This transformation leads to the integral being rewritten as $$\int_{1}^{\infty} \frac{\{t\}}{t^{2}} \, dt$$. The proof involves evaluating this integral through a series of steps that culminate in the limit of a logarithmic expression. Ultimately, the result confirms that the integral equals $1 - \gamma$. The discussion effectively demonstrates the relationship between the integral and the Euler-Mascheroni constant, $\gamma$.
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Prove the following
$$\int^1_0 \left \{ \frac{1}{x} \right \} \, dx = 1- \gamma $$​
 
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ZaidAlyafey said:
Prove the following
$$\int^1_0 \left \{ \frac{1}{x} \right \} \, dx = 1- \gamma $$​

[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

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