Recall that $x$ is a fixed element of $]0;1[$. Establish the identity
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { n + x } + \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { n + 1 - x }$$