integral representations of the Mascheroni constant

Mascheroni’s constant can be expressed by the following integrals:

$\gamma$$=$$-{\displaystyle {\int }_0^1}\log (-\log x)𝑑x$

$\gamma$$=$$-{\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-x}\log xdx$

$\gamma$$=$${\displaystyle {\int }_0^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{e^t-1}}-{\displaystyle \frac{1}{t{e}^t}}\right)𝑑t$

$\gamma$$=$${\displaystyle {\int }_0^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{t}}-{\displaystyle \frac{1}{1+t}}-{\displaystyle \frac{1}{t{e}^t}}\right)𝑑t$